Influence of Radial Basis Activation Functions on Intelligent Controller for Robotic Manipulators

Technical Summary: Influence of Radial Basis Activation Functions on Intelligent Controller for Robotic Manipulators

Problem Statement
Robotic manipulator trajectory tracking is frequently addressed using model-based nonlinear control methods, such as feedback linearization. While effective under nominal conditions, these approaches suffer performance degradation due to parametric uncertainties, friction, unmodeled dynamics, and external disturbances. Although adaptive and learning-based extensions exist to enhance robustness, the specific influence of the activation function (kernel) within Radial Basis Function (RBF) neural network disturbance estimators on closed-loop behavior remains under-explored. While Lyapunov stability is generally guaranteed for bounded activation functions, the impact of kernel shape on transient response, steady-state accuracy, and control smoothness is not systematically characterized.

Methodology
The authors propose an intelligent control framework that integrates feedback linearization with an online neural disturbance estimator based on RBF networks.

• Control Structure: The controller utilizes a model-based feedback linearization law augmented by an adaptive term ($\hat{d}$) to compensate for total uncertainty ($d$). The disturbance estimator is defined as $\hat{d}_i = w_i^\top \phi_i(s_i)$, where $w_i$ represents adaptive weights and $\phi_i$ represents the activation functions.
• Stability Analysis: A Lyapunov-based adaptation law is employed to update the neural weights. To ensure the boundedness of closed-loop signals and prevent weight drift, a projection algorithm is applied, guaranteeing that weights remain within a convex region. This ensures the tracking error converges to a compact region around the origin.
• Experimental Setup: The framework was implemented on a single-degree-of-freedom (DOF) Quanser QArm robotic manipulator within a digital twin environment (Quanser Interactive Labs). The system was tested under hardware-timed acquisition (500 Hz) with three distinct reference trajectories: sinusoidal, square (step-like), and triangular (ramp-like).
• Comparative Protocol: To isolate the effect of the activation function, all other parameters were held constant across tests, including controller gains ($\lambda_i$), learning rates ($\eta_i$), projection bounds ($\mu_i$), and the number of neurons ($n_i = 6$). Three specific kernels were evaluated:
  1. Gaussian: $\kappa_G(\rho) = \exp(-\rho^2)$ (Local support, baseline).
  2. Laplacian: $\kappa_L(\rho) = \exp(-|\rho|)$ (Local support).
  3. Inverse Multiquadratic (IMQ): $\kappa_{IMQ}(\rho) = 1/\sqrt{1+\rho^2}$ (Global support).

Key Results
Experimental results quantified via Root-Mean-Square Error (RMSe), Integral of Absolute Error (IAE), and control effort (RMSPWM) revealed distinct performance trade-offs based on kernel selection:

• Continuous Trajectories (Sinusoidal and Triangular): The Inverse Multiquadratic (IMQ) kernel demonstrated superior performance. Compared to the baseline (conventional feedback linearization without neural compensation), the IMQ kernel reduced RMSe by 51.9% for sinusoidal waves and 49.0% for triangular waves. The Laplacian and Gaussian kernels also improved performance over the baseline but to a lesser extent.
• Discontinuous Trajectories (Square Reference): A critical trade-off emerged. While all neural kernels reduced the IAE (indicating better overall error accumulation), they introduced transient overshoot and increased settling times compared to the baseline. Specifically, the IMQ kernel resulted in the highest overshoot (6.86%) and settling time (2.624 s), while the baseline controller maintained zero overshoot.
• Control Effort: The increase in control effort (RMSPWM) was minimal across all intelligent controllers, generally remaining below a 3% increase relative to the baseline, despite significant improvements in tracking accuracy.

Significance and Claims
The paper asserts that the choice of activation function in RBF-based intelligent controllers should be viewed not merely as an implementation detail, but as a structural design parameter. The findings indicate that:

1. Kernel Support Matters: Kernels with global support (like IMQ) provide more holistic approximation of disturbance manifolds for smooth, continuous trajectories, leading to higher steady-state precision.
2. Transient vs. Steady-State Trade-off: Localized kernels (Gaussian, Laplacian) or the baseline controller may be preferable for applications requiring strictly monotonic step responses, as global kernels can induce transient aggressiveness under discontinuous commands.
3. Design Flexibility: Engineers can tune the activation function to prioritize either maximum steady-state precision (favoring IMQ for smooth paths) or minimal transient overshoot (favoring localized kernels or baseline for step inputs), all while maintaining the theoretical stability guarantees provided by the Lyapunov-based adaptation law.

The study concludes that while stability is preserved for all bounded kernels, the selection of the kernel directly shapes adaptation dynamics and practical closed-loop performance, offering a mechanism to tailor intelligent controllers to specific trajectory requirements.