In the realm of advanced robot technology, collaborative robots have emerged as pivotal systems due to their ability to perform complex tasks with high flexibility. These robots often incorporate redundant degrees of freedom, enabling them to adapt to dynamic environments beyond the minimum requirements for specific operations. However, this flexibility introduces challenges, such as joint velocity fluctuations under varying loads, which can compromise control precision and stability. As a key aspect of modern robot technology, addressing these fluctuations is essential for enhancing performance in applications like manufacturing, healthcare, and automation. In this article, I present a comprehensive method for suppressing joint fluctuations in collaborative robots, leveraging adaptive algorithms and disturbance observers to mitigate speed variations. This approach not only improves control accuracy but also underscores the evolution of robot technology in handling nonlinear dynamics. Throughout this discussion, I will emphasize the integration of mathematical models and experimental validations, using tables and equations to summarize key concepts, thereby contributing to the broader field of robot technology.
The foundation of this method lies in analyzing the joint motion state of collaborative robots. Typically, joint modules in such robots are driven by dual-stator permanent magnet brushless torque motors, where inner and outer stators are isolated by magnetic rings to form independent circuits. This design allows for separate control of torque distribution, which is crucial for suppressing velocity fluctuations. The motion equation of the joint drive motor can be expressed as:
$$ \frac{d\omega}{dt} = \frac{1}{J} (T_e – B\omega – T_l) $$
where \( \omega \) represents the angular velocity, \( J \) is the rotor inertia, \( T_e \) denotes the electromagnetic torque, \( B \) is the friction coefficient, and \( T_l \) is the load torque. The electromagnetic torque is further defined as:
$$ T_e = 1.5p \left[ \lambda i_q + (L_d – L_q) i_d i_q \right] $$
Here, \( p \) is the number of pole pairs, \( \lambda \) is the equivalent flux linkage, \( i_d \) and \( i_q \) are the d-axis and q-axis currents, and \( L_d \) and \( L_q \) are the stator inductances. This equation highlights how magnetic fields and currents influence torque, forming the basis for controlling velocity fluctuations in robot technology applications. To monitor these parameters in real-time, a full-dimensional state observer is designed, incorporating both inner and outer loops to track joint speed variations. The observer structure feedbacks speed data, enabling precise detection of fluctuations. The state equation for the motor can be written in matrix form:
$$ \begin{bmatrix} \dot{i}_d \\ \dot{i}_q \\ \dot{\omega} \end{bmatrix} = \begin{bmatrix} -\frac{R}{L} & p\omega & 0 \\ -p\omega & -\frac{R}{L} & -\frac{p\lambda}{L} \\ 0 & \frac{1.5p\lambda}{J} & 0 \end{bmatrix} \begin{bmatrix} i_d \\ i_q \\ \omega \end{bmatrix} + \begin{bmatrix} \frac{u_d}{L} \\ \frac{u_q}{L} \\ -\frac{T_l}{J} \end{bmatrix} $$
where \( R \) is the winding resistance, \( L \) is the equivalent inductance, and \( u_d \) and \( u_q \) are the d-axis and q-axis voltages. By defining a characteristic polynomial for the observer, such as \( f(s) = \det(sI – (A – LC)) \), where \( A \) is the system matrix, \( L \) is the feedback gain matrix, and \( C \) is the output matrix, a closed-loop observer system is constructed:
$$ \dot{\hat{x}} = A\hat{x} + Bu + L(y – \hat{y}), \quad \hat{y} = C\hat{x} $$
This observer provides estimated state variables, including rotor position angle \( \theta \), mechanical angular velocity \( \gamma \), and load torque \( T_l \), which are critical for feedback control in robot technology systems. The integration of this observer facilitates the development of a robust control architecture, as detailed in the following sections.
To address velocity fluctuations, a control speed feedback channel is established using a PI controller, forming the inner loop compensation回路. This involves allocating different torque values to the inner and outer stators for feedforward compensation, which preliminarily suppresses joint control fluctuations. The control system框图 includes inputs for reference speed and actual motor speed, processed by the PI controller to generate outputs for disturbance observers and torque distributors. The output of the PI controller can be represented as:
$$ u = K_p e + K_i \int e \, dt $$
where \( e \) is the speed error, \( K_p \) is the proportional gain, and \( K_i \) is the integral gain. This setup helps in mitigating periodic and nonlinear disturbances, which are common in robot technology applications. For instance, periodic disturbances can be modeled using Fourier series analysis:
$$ \delta_p = \sum_{i=1}^{5} \left( a_i \sin(\omega_i t) + b_i \cos(\omega_i t) \right) T_l $$
where \( a_i \) and \( b_i \) are control parameters dependent on current conditions, and \( \omega_i \) represents the actual joint speed. By combining this with nonlinear disturbances, an appropriate feedforward compensation value is derived, enhancing the suppression of joint fluctuations. The table below summarizes key parameters used in the inner loop compensation:
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Proportional Gain | \( K_p \) | 1.2 | — |
| Integral Gain | \( K_i \) | 0.5 | — |
| Resistance | \( R \) | 0.1 | Ω |
| Inductance | \( L \) | 0.05 | H |
Building on the inner loop, an outer loop compensation strategy is implemented using an adaptive disturbance estimator to account for transmission errors. This estimator accurately calculates periodic and nonlinear disturbances, providing specific feedforward compensation values. The adaptive algorithm involves estimating disturbances as:
$$ \hat{\delta}_i = \frac{r \omega^{-1} \Phi \tau_i \nu}{\nu^2 + r \omega^{-1} \Phi \tau_i \nu + \omega_i^2} + \delta (\omega^{-1} \Phi – 1) \zeta $$
where \( \tau \) is the adaptation rate, \( \nu \) is a variable in the transfer function, \( \Phi \) is the actual system transfer function, \( \omega \) is the nominal transfer function, and \( \zeta \) is the output of the speed闭环 PI controller. The total disturbance is then used to allocate torque between the inner and outer stators via a torque distributor, minimizing the objective function:
$$ \min \sum_{i=1}^{5} (a_i^2 + b_i^2 + \sigma_2^2) $$
where \( \sigma_1 \) and \( \sigma_2 \) represent the allocated torques for the inner and outer stators, respectively, ensuring that the inner stator current remains unsaturated. This comprehensive approach effectively suppresses joint velocity fluctuations, showcasing the advancements in robot technology for handling complex dynamics. The integration of adaptive algorithms and disturbance observers not only improves robustness but also aligns with the ongoing innovations in robot technology.
To validate the proposed method, an experimental setup was designed using a collaborative robot with seven degrees of freedom, each joint equipped with a dual-stator permanent magnet synchronous motor, a harmonic reducer, and a controller. This setup reflects typical configurations in robot technology, allowing for testing under various speed conditions. The motors were characterized by parameters such as winding resistance, inductance, and rated torque, as detailed in the table below:
| Parameter | Inner Stator | Outer Stator | Unit |
|---|---|---|---|
| Winding Resistance | 0.09 | 0.17 | Ω |
| Winding Inductance | 0.008 | 0.079 | H |
| Rated Speed | 1000 | 2000 | r/min |
| Rated Torque | 0.09 | 0.27 | Nm |
| Current Torque Constant | 0.013 | 0.045 | Nm/A |
In the experiments, joint speed responses were monitored under low-speed and high-speed conditions using the state observer. The results indicated significant fluctuations without suppression, with amplitudes increasing at higher speeds. For example, the speed response curve showed variations up to 0.6 rad/s in uncontrolled scenarios. After applying the proposed method, the maximum fluctuation amplitudes were reduced to 0.05 rad/s for low-speed and 0.07 rad/s for high-speed conditions, demonstrating the efficacy of this approach in robot technology. The root mean square (RMS) values of steady-state errors were calculated using:
$$ \text{RMS} = \sqrt{\frac{1}{N} \sum_{t=1}^{N} e_t^2 } $$
where \( N \) is the total operation time and \( e_t \) is the speed error at time \( t \). The RMS values remained below 0.5 across all tests, highlighting the method’s stability and reliability. Comparative analysis with alternative methods revealed that the proposed technique outperformed others in suppressing both low-frequency and mid-high-frequency fluctuations, thanks to its adaptive disturbance estimation and torque distribution mechanisms. This aligns with the goals of robot technology to achieve precise control under dynamic loads.
In conclusion, the joint fluctuation suppression method presented here leverages state observers, adaptive algorithms, and disturbance estimators to enhance the performance of collaborative robots. By addressing speed variations through inner and outer loop compensations, this approach minimizes control deviations and improves stability, contributing significantly to the field of robot technology. The experimental results confirm its effectiveness, with low fluctuation amplitudes and RMS values, making it suitable for real-world applications. As robot technology continues to evolve, such methods will play a crucial role in enabling robots to perform complex tasks with higher accuracy and efficiency. Future work could focus on extending this approach to multi-robot systems or integrating machine learning for adaptive parameter tuning, further advancing the capabilities of robot technology.