In the field of industrial automation, robot technology has revolutionized manufacturing processes, particularly in tasks requiring precise force control. As a researcher focused on enhancing robotic capabilities, I have observed that complex surface grinding operations often suffer from force fluctuations due to uncertainties in workpiece curvature, leading to compromised surface quality. Traditional position-based control methods fall short in such contact-based tasks, as they cannot adapt to dynamic environmental changes. This article presents a novel force control strategy that integrates a variable integral PID feedforward controller with an adaptive control algorithm, specifically designed to improve force tracking in robot technology applications. By leveraging real-time force and position data, this method dynamically adjusts robot motion offsets, mitigating errors caused by unknown工况 parameters. Through simulations and experimental validation, I demonstrate that this approach significantly reduces force tracking errors and steady-state adjustment times, addressing the limitations of conventional impedance control. The integration of this advanced control scheme exemplifies how robot technology can evolve to handle complex manufacturing challenges, ensuring higher precision and consistency in grinding processes.
The core of this work lies in addressing the steady-state errors inherent in traditional position-based impedance control. In robot technology, impedance control models the interaction between the robot end-effector and the environment as a mass-spring-damper system. The standard impedance model in one dimension is given by:
$$ m_d(\ddot{x} – \ddot{x}_r) + b_d(\dot{x} – \dot{x}_r) + k_d(x – x_r) = f_d – f_e $$
where \( m_d \), \( b_d \), and \( k_d \) are the desired inertia, damping, and stiffness coefficients of the robot impedance model, \( x_r \) is the reference position, \( x \) is the actual robot end-effector position, \( f_d \) is the desired contact force, and \( f_e \) is the actual contact force. The environment is often modeled as a spring-damper system:
$$ f_e = b_e \dot{x} + k_e (x – x_e) $$
where \( k_e \) and \( b_e \) are the environmental stiffness and damping coefficients, and \( x_e \) is the environmental position. Analyzing the steady-state behavior, the force tracking error \( \Delta f = f_d – f_e \) under step inputs reveals a fundamental limitation:
$$ \Delta f_{ss} = \lim_{s \to 0} s \Delta f(s) = \frac{k_d k_e}{k_d + k_e} \left( \frac{f_d}{k_e} + x_e – x_r \right) $$
This equation shows that the steady-state error depends on the impedance parameters and the accuracy of the reference trajectory. If the environmental parameters \( k_e \), \( b_e \), and \( x_e \) are unknown or inaccurate, as is common in real-world robot technology applications, the force tracking performance degrades. To achieve zero steady-state error, one could set \( k_d = 0 \), but this reduces system robustness and is impractical for low-stiffness end-effectors. Alternatively, modifying the reference trajectory to \( x_r = f_d / k_e + x_e \) can eliminate the error, but this requires accurate knowledge of environmental parameters, which is often unavailable. This insight motivates the development of an adaptive control approach that estimates these parameters online, enhancing the adaptability of robot technology in uncertain environments.
The proposed adaptive variable integral impedance control method combines a variable integral PID controller with an adaptive law to compensate for uncertainties. The variable integral PID component serves as a feedforward controller, improving dynamic response and reducing overshoot. The PID-impedance control equation is formulated as:
$$ m_d(\ddot{x} – \ddot{x}_r) + b_d(\dot{x} – \dot{x}_r) + k_d(x – x_r) = k_P \Delta f + k_I \int_0^t \Delta f \, dt + k_D \dot{\Delta f} $$
where \( k_P \), \( k_I \), and \( k_D \) are the PID gains. To address integral windup and improve precision, the variable integral algorithm adjusts the integral action based on the error magnitude. The integral term is computed as:
$$ u_i(k) = k_I \left( \sum_{i=0}^{k-1} e(i) + f(e(k)) e(k) \right) T $$
where \( e(k) \) is the error at step \( k \), \( T \) is the sampling time, and \( f(e(k)) \) is a function that varies between 0 and 1:
$$ f(e(k)) = \begin{cases}
1 & \text{if } |e(k)| \leq B \\
\frac{A – |e(k)| + B}{A} & \text{if } B < |e(k)| \leq A + B \\
0 & \text{if } |e(k)| > A + B
\end{cases} $$
Here, \( A \) and \( B \) are constants that define the integration behavior. This approach ensures that integration is suppressed when errors are large, preventing overshoot, and enhanced when errors are small, reducing steady-state error. The complete variable integral PID control law is:
$$ u(k) = k_P e(k) + k_I \left( \sum_{i=0}^{k-1} e(i) + f(e(k)) e(k) \right) T + k_D (e(k) – e(k-1)) $$
This modification significantly improves the force tracking performance in robot technology by providing a smoother and more responsive control action.
The adaptive control component estimates the environmental parameters \( \hat{x}_e \), \( \hat{k}_e \), and \( \hat{b}_e \) online using a Lyapunov-based approach. The estimated reference trajectory is:
$$ \hat{x}_r = \frac{f_d}{\hat{k}_e} + \hat{x}_e $$
and the estimated contact force is:
$$ \hat{f}_e = \hat{b}_e \dot{x} + \hat{k}_e x – \hat{k}_e \hat{x}_e $$
The force estimation error \( e_f = \hat{f}_e – f_e \) can be expressed as \( e_f = \phi \theta \), where \( \phi = [\dot{x}, x, -1] \) and \( \theta = [\hat{b}_e – b_e, \hat{k}_e – k_e, \hat{k}_e \hat{x}_e – k_e x_e]^T \). To ensure stability, a Lyapunov function \( V = \theta^T \Lambda \theta \) is defined, with \( \Lambda \) being a positive definite diagonal matrix. The parameter update laws are derived as:
$$ \hat{b}_e(k+1) = \hat{b}_e(k) – \frac{\dot{x}_{k+1} e_f(k)}{\lambda_1} $$
$$ \hat{k}_e(k+1) = \hat{k}_e(k) – \frac{x_{k+1} e_f(k)}{\lambda_2} $$
$$ \hat{x}_e(k+1) = \hat{x}_e(k) + \frac{e_f(k)}{\hat{k}_e(k+1)} \left( \frac{x_{k+1} \hat{x}_e(k)}{\lambda_2} + \frac{1}{\lambda_3} \right) $$
where \( \lambda_1 \), \( \lambda_2 \), and \( \lambda_3 \) are positive constants. These updates ensure that \( \hat{f}_e \to f_e \) over time, leading to \( f_e \to f_d \), thus achieving zero force tracking error. This adaptive mechanism is crucial for robot technology to handle unknown or varying environmental conditions, such as changes in workpiece curvature or stiffness during grinding.
To validate the proposed method, I conducted extensive simulations comparing traditional impedance control with the adaptive variable integral approach under various scenarios. The simulations were performed in MATLAB, modeling the robot-environment interaction. The impedance parameters were set as \( m_d = 1 \), \( b_d = 10 \), and \( k_d = 100 \), while the PID gains were \( k_P = 10 \), \( k_I = 100 \), \( k_D = 0.4 \), with \( A = 0.4 \) and \( B = 0.6 \) for the variable integral function. The adaptive parameters were \( \lambda_1 = 1000 \), \( \lambda_2 = 100 \), and \( \lambda_3 = 0.55 \). The desired force \( f_d \) was set to 10 N. Four test cases were evaluated: fixed position and stiffness, step change in position, step change in stiffness, and sinusoidal variation in position. The results are summarized in the table below, highlighting key performance metrics such as overshoot, settling time, and steady-state error.
| Test Case | Control Method | Position Overshoot (mm) | Force Overshoot (N) | Settling Time (s) | Steady-State Error (N) |
|---|---|---|---|---|---|
| Fixed Position and Stiffness | Traditional | 0.339 | 12.062 | 0.74 | 0.035 |
| Fixed Position and Stiffness | Adaptive | 0.320 | 11.338 | 0.42 | 0.0057 |
| Step Position Change | Traditional | 0.420 | 11.031 | 4.28 | N/A |
| Step Position Change | Adaptive | 0.410 | 10.673 | 3.41 | N/A |
| Step Stiffness Change | Traditional | 0.207 | 8.562 (min) | 4.41 | N/A |
| Step Stiffness Change | Adaptive | 0.220 | 9.050 (min) | 3.34 | N/A |
| Sinusoidal Position | Traditional | N/A | 8.549 – 11.348 | N/A | N/A |
| Sinusoidal Position | Adaptive | N/A | 9.761 – 10.974 | N/A | N/A |
In the fixed position and stiffness case, the adaptive control reduced position overshoot by 48.69%, force overshoot by 35.11%, settling time by 43.24%, and steady-state error by 83.71%. For step changes, the adaptive method achieved faster recovery to steady-state, demonstrating its superiority in handling dynamic uncertainties. Under sinusoidal position variations, the adaptive control confined force fluctuations to a narrower range (9.761–10.974 N) compared to traditional control (8.549–11.348 N), indicating better tracking performance. These simulations underscore the potential of this method in advancing robot technology for precision tasks.
To further assess practical applicability, I implemented the control strategy on an industrial robot grinding system, consisting of a KUKA KR70 R2100 robot, an ATI 9105-NET Delta IP60 force sensor, and a custom software interface for trajectory planning and force monitoring. The robot control system operated at a 4 ms cycle, with force sampling at 7000 Hz. A propeller blade was used as the workpiece, and grinding experiments were conducted with a desired force of 4 N in the Z-direction (normal to the surface). The traditional impedance control and adaptive variable integral impedance control were compared in terms of force tracking and surface roughness. The force data during grinding is illustrated in the following table, which shows the range of force variations and corresponding surface roughness measurements at different locations on the blade.
| Control Method | Force Range (N) | Average Surface Roughness (µm) | Notes |
|---|---|---|---|
| Traditional Impedance | 4.194 ± 1.506 | 0.65 (0.81 in damaged areas) | Significant fluctuations and undersampling |
| Adaptive Variable Integral | 4.049 ± 0.972 | 0.34 (0.42 after repair) | Stable force tracking and consistent quality |
The experimental results confirm that the adaptive control method maintains force within a tighter range (4.049 ± 0.972 N) compared to traditional control (4.194 ± 1.506 N), leading to improved surface quality. The average roughness decreased from approximately 0.65 µm to 0.34 µm, with even damaged areas showing better recovery (0.42 µm after repair). This enhancement is attributed to the adaptive controller’s ability to estimate environmental parameters and adjust the reference trajectory in real-time, reducing the impact of positioning errors and workpiece variations. The success of these experiments highlights how integrating advanced control algorithms can elevate the capabilities of robot technology in demanding applications like grinding.
In conclusion, the adaptive variable integral impedance control method presented here addresses critical challenges in robot technology for force-controlled grinding. By combining a variable integral PID feedforward controller with an adaptive parameter estimation scheme, this approach minimizes force tracking errors, reduces settling time, and improves surface quality. The simulations and experiments consistently demonstrate its superiority over traditional methods, particularly in handling unknown or varying environmental conditions. As robot technology continues to evolve, such adaptive control strategies will play a pivotal role in enabling robots to perform complex tasks with higher precision and autonomy. Future work will focus on vibration suppression during grinding and extending this method to multi-axis force control scenarios, further pushing the boundaries of what robot technology can achieve in industrial automation.