In the rapidly evolving field of robot technology, underactuated systems have garnered significant attention due to their cost-effectiveness, lightweight design, and adaptability in complex environments such as space and deep-sea exploration. These systems, characterized by having fewer control inputs than degrees of freedom, present unique challenges in achieving stable and precise control. This article addresses the stabilization issues in planar two-degree-of-freedom (DoF) underactuated robots, which are fundamental models in robot technology, by proposing a unified control strategy based on trajectory planning and tracking. The approach integrates insights from both planar Acrobot and Pendubot structures, leveraging their coupling constraints to enable simultaneous control of actuated and underactuated links. Through extensive simulation and analysis, this strategy demonstrates superior performance in reducing control time and torque while maintaining system stability, contributing to the advancement of robot technology in practical applications.
The core of this work lies in developing a generalized framework for planar underactuated robots, which are pivotal in robot technology for tasks requiring efficient resource utilization. Underactuated robots, such as those with a single passive joint, often face instability during motion due to the inherent complexities of their dynamics. By focusing on planar two-DoF systems, this research provides a foundation for extending control strategies to more complex multi-DoF systems in robot technology. The unified control strategy not only bridges the gap between different structural configurations but also enhances the reliability and performance of robotic systems in real-world scenarios.
System Modeling and Underactuation Analysis in Robot Technology
In robot technology, accurate dynamic modeling is essential for designing effective control strategies. The planar two-DoF underactuated robot consists of two links, with parameters including mass, length, angle, moment of inertia, and control torque. The system can be classified into two types based on the position of the passive joint: planar Acrobot (first joint passive) and planar Pendubot (second joint passive). Despite their structural differences, both share common dynamic properties that can be captured in a unified model.
The dynamic equations of motion are derived using Lagrangian mechanics, resulting in the following matrix form:
$$ M(q) \ddot{q} + H(q, \dot{q}) = \tau $$
where \( q = [q_1, q_2]^T \) represents the joint angles, \( \dot{q} \) and \( \ddot{q} \) are the angular velocities and accelerations, respectively, \( M(q) \in \mathbb{R}^{2 \times 2} \) is the symmetric and positive-definite inertia matrix, \( H(q, \dot{q}) \in \mathbb{R}^{2 \times 1} \) combines Coriolis and centrifugal forces, and \( \tau \in \mathbb{R}^{2 \times 1} \) is the control torque vector. For planar Acrobot, \( \tau = [0, \tau_2]^T \), and for planar Pendubot, \( \tau = [\tau_1, 0]^T \). The elements of \( M(q) \) and \( H(q, \dot{q}) \) are given by:
$$ M(q) = \begin{bmatrix} m_1 + m_2 & -m_2 l_2 \sin q_2 \\ -m_2 l_2 \sin q_2 & m_2 l_2^2 + J_2 \end{bmatrix} $$
$$ H(q, \dot{q}) = \begin{bmatrix} -m_2 l_2 \dot{q}_2^2 \cos q_2 \\ 0 \end{bmatrix} $$
The underactuation constraint, which arises from the passive joint, is expressed as:
$$ M_{uu} \ddot{q}_u + M_{ua} \ddot{q}_a + H_u = 0 $$
where \( \ddot{q}_a \) and \( \ddot{q}_u \) denote the accelerations of the actuated and underactuated links, respectively. This constraint highlights the coupling between the links, a critical aspect in robot technology for indirect control. By integrating this constraint into the trajectory planning, the underactuated link’s state can be regulated through the actuated link’s motion.
To further illustrate the system parameters, Table 1 summarizes the key variables used in the model, which are common in robot technology applications.
| Parameter | Description | Value |
|---|---|---|
| \( m_i \) | Mass of link i | 1.0 kg |
| \( L_i \) | Length of link i | 1.0 m |
| \( l_i \) | Distance from joint to center of mass | 0.5 m |
| \( J_i \) | Moment of inertia | 0.083 kg·m² |
| \( q_i \) | Joint angle | Variable |
| \( \tau_i \) | Control torque | Variable |
The coupling relationship enables the derivation of the underactuated link’s state over time:
$$ \dot{q}_u(t_s) = -\int_0^T \frac{M_{ua} \ddot{q}_a + H_u}{M_{uu}} \, dt + \dot{q}_{u0} $$
$$ q_u(t_s) = \int_0^T \dot{q}_u \, dt + q_{u0} $$
where \( t_s \) is the current time, \( T \) is the total control time, and \( q_{u0} \) and \( \dot{q}_{u0} \) are the initial angle and angular velocity of the underactuated link. This formulation is pivotal in robot technology for designing controllers that account for dynamic interactions.
Trajectory Planning and Tracking Control for Enhanced Robot Technology
In robot technology, trajectory planning is a fundamental step for achieving desired motion profiles while maintaining stability. The proposed control strategy involves a two-stage trajectory for the actuated link, ensuring that both links reach their target states simultaneously. The first stage, \( T_{a1}(t) \), drives the actuated link quickly to its target position, while the second stage, \( T_{a2}(t) \), adjusts the underactuated link’s state through optimized parameters.
The first-stage trajectory is designed as:
$$ T_{a1}(t) = \begin{cases} q_{a0} + \dot{q}_r \left( \frac{t}{t_f} – \frac{1}{2\pi} \sin \alpha \right), & 0 \leq t \leq t_f \\ q_{ad}, & t > t_f \end{cases} $$
where \( q_{a0} \) is the initial angle, \( q_{ad} \) is the target angle, \( \dot{q}_r = q_{ad} – q_{a0} \), \( \alpha = 2\pi t / t_f \), and \( t_f \) is the time to reach the target. The derivatives of \( T_{a1}(t) \) are continuous, ensuring smooth torque profiles:
$$ \dot{T}_{a1}(t) = \begin{cases} \frac{\dot{q}_r}{t_f} (1 – \cos \alpha), & 0 \leq t \leq t_f \\ 0, & t > t_f \end{cases} $$
$$ \ddot{T}_{a1}(t) = \begin{cases} \frac{\dot{q}_r}{t_f^2} (2\pi \sin \alpha), & 0 \leq t \leq t_f \\ 0, & t > t_f \end{cases} $$
The second-stage trajectory, \( T_{a2}(t) \), incorporates adjustable parameters to fine-tune the underactuated link’s behavior:
$$ T_{a2}(t) = A_{11} \text{sech}(\alpha_1) + A_{12} \text{sech}(\alpha_2) $$
with
$$ \alpha_1 = \frac{\lambda (2t – t_{m1})}{t_{m1}}, \quad \alpha_2 = \frac{\lambda (2t – t_f – t_{m1})}{t_f – t_{m1}} $$
where \( \lambda \geq \rho \) (a constant), \( t_{m1} \in (0, t_f) \) is a connection time, and \( A_{11} \), \( A_{12} \) are amplitudes. The derivatives of \( T_{a2}(t) \) are also continuous, preventing torque discontinuities that could destabilize the system in robot technology applications.
The combined trajectory \( T_a = T_{a1} + T_{a2} \) ensures that the actuated link tracks a smooth path while indirectly controlling the underactuated link. This approach is crucial in robot technology for achieving coordinated motion without explicit control of the passive joint.
To implement the tracking control, a sliding mode controller is designed for the actuated link. Define the state vector \( x = [q_1, q_2, \dot{q}_1, \dot{q}_2]^T \). The state-space representation is:
$$ \dot{x} = f_a(x) + g_{aa}(x) \tau_a $$
where \( f_a(x) = -M^{-1}(q) H(q, \dot{q}) \) and \( g_{aa}(x) = -M^{-1}(q) \). The sliding surface \( S_a \) is constructed as:
$$ S_a = \mu_a e_a(t) + \dot{e}_a(t) $$
with \( e_a(t) = T_{ad} – T_a(t) \) and \( \dot{e}_a(t) = \dot{T}_{ad} – \dot{T}_a(t) \), where \( \mu_a > 0 \) is a constant. The control torque \( \tau_a \) is derived to enforce sliding mode dynamics:
$$ \tau_a = \frac{-\zeta_a S_a – \xi_a \text{sgn}(S_a) – \mu_a \dot{e}_a – f_a(x) + \ddot{T}_a}{g_{aa}(x)} $$
where \( \zeta_a \) and \( \xi_a \) are positive constants. The Lyapunov function \( V = \frac{1}{2} S_a^2 \) ensures stability, as \( \dot{V} \leq 0 \), guaranteeing convergence of the tracking error. This controller is integral to robot technology for robust performance in the presence of uncertainties.
Parameter Optimization Using Intelligent Algorithms in Robot Technology
In robot technology, optimization algorithms play a vital role in tuning control parameters for enhanced performance. The differential evolution (DE) algorithm is employed to optimize the trajectory parameters \( A_{11} \), \( A_{12} \), \( t_{m1} \), and \( t_f \), minimizing an evaluation function based on the underactuated link’s state error.
The evaluation function is defined as:
$$ h = \left| q_u(t_f) – q_{ud} \right| + \left| \dot{q}_u(t_f) \right| $$
where \( q_{ud} \) is the target angle for the underactuated link. The DE algorithm iteratively updates the parameters through mutation, crossover, and selection steps until \( h \leq \varepsilon_1 \), a small threshold. This process ensures that the optimized trajectory enables simultaneous control of both links.
Table 2 outlines the DE parameters used in the optimization, which are standard in robot technology for global optimization.
| Parameter | Description | Value |
|---|---|---|
| \( N \) | Population size | 40 |
| \( p_m \) | Mutation probability | 0.3 |
| \( p_c \) | Crossover probability | 0.7 |
| \( G_{\text{max}} \) | Maximum generations | 200 |
| \( \varepsilon \) | Convergence threshold | 0.0005 |
The optimization results for planar Acrobot and Pendubot are summarized in Table 3, demonstrating the effectiveness of the approach in robot technology.
| System | \( A_{11} \) | \( A_{12} \) | \( t_{m1} \) (s) | \( t_f \) (s) |
|---|---|---|---|---|
| Planar Acrobot | 0.379 | -0.936 | 2.245 | 7.669 |
| Planar Pendubot | -0.194 | -0.066 | 5.808 | 11.862 |
By leveraging DE, the control strategy achieves minimal state errors and smooth torque profiles, which are essential for reliable robot technology implementations.
Simulation Results and Analysis in Robot Technology
Simulations were conducted in MATLAB/Simulink to validate the unified control strategy, using the parameters from Table 1. The results for planar Acrobot and Pendubot are compared with existing methods, highlighting the advantages in robot technology.
For planar Acrobot, the initial angles are \( q_1 = 0 \), \( q_2 = 0 \), and target angles are \( q_1 = -2.0 \) rad, \( q_2 = 14.1 \) rad. The optimized parameters from Table 3 yield a control time of \( t_f = 7.669 \) s. The angles and angular velocities converge smoothly to their targets, with velocities maintained within ±5 rad/s and control torque \( \tau_2 \) within ±4 N·m. Compared to prior approaches, this strategy reduces control time and torque fluctuations, enhancing stability in robot technology.
For planar Pendubot, initial angles are \( q_1 = 0 \), \( q_2 = 0 \), and target angles are \( q_1 = 0.5 \) rad, \( q_2 = -1.0 \) rad. With \( t_f = 11.862 \) s, the angles reach their targets, angular velocities stay within ±1 rad/s, and torque \( \tau_1 \) is within ±1.5 N·m. The torque profile is continuous and smooth, outperforming existing methods in robot technology by providing better stability and lower energy consumption.
Table 4 provides a comparative analysis of the proposed strategy against literature methods, emphasizing its benefits in robot technology.
| Metric | Existing Methods | Proposed Strategy |
|---|---|---|
| Average Angular Velocity | ±5 rad/s or higher | ±3 rad/s |
| Average Control Torque | ±4 N·m or higher | ±2.75 N·m |
| Stabilization Time | ~10 s | ~9.5 s (average) |
| Torque Continuity | Discontinuous in some cases | Smooth and continuous |
The simulations confirm that the unified control strategy effectively handles both planar Acrobot and Pendubot structures, demonstrating its versatility in robot technology. The integration of trajectory planning, tracking control, and optimization ensures robust performance, making it suitable for real-world applications where reliability and efficiency are paramount.
Conclusion and Future Directions in Robot Technology
This article presents a unified control strategy for planar two-DoF underactuated robots, leveraging trajectory planning and tracking to achieve simultaneous stabilization of actuated and underactuated links. The approach addresses the limitations of existing methods by incorporating a two-stage trajectory optimized through differential evolution, resulting in reduced control time, lower torque, and smooth motion profiles. The strategy’s effectiveness is validated through simulations, showing significant improvements in performance metrics relevant to robot technology.
The implications of this work extend beyond two-DoF systems, providing a foundation for controlling more complex underactuated robots in advanced robot technology. Future research will focus on extending the strategy to multi-DoF systems, incorporating adaptive control for handling uncertainties, and applying it to real-world robotic platforms in dynamic environments. By continuing to refine these techniques, robot technology can achieve greater autonomy and reliability, enabling broader adoption in critical fields such as aerospace, marine exploration, and industrial automation.
In summary, the unified control strategy represents a significant step forward in robot technology, offering a scalable and efficient solution for underactuated system control. Its ability to generalize across different structures while maintaining high performance underscores its potential to drive innovation in the field, paving the way for next-generation robotic systems.