We define the trajectory-tracking error as e=q-qd and construct a linear sliding surface: 4s=e˙+Λe, where Λ=diag{λ1,…,λn} is a positive definite diagonal matrix. Differentiating Eq. (4) and substituting into Eq. (2) yields the sliding-surface dynamics: 5M0(q)s˙=M0(q)(q¨-q¨d+Λe˙)=τ+f(x)+τd-C0q˙-G0-M0(q)(q¨d-Λe˙). To achieve high-precision tracking and eliminate chattering, an RBF-based super-twisting sliding-mode control (RBF-STSMC) is designed: 6τ=M0(q)(q¨d-Λe˙)+C0(q,q˙)q˙+G0(q)-W^TΦ(x)+M0(q)ust, where W^∈RN×n is the estimated weight matrix of the RBF neural network, used for feedforward compensation of system uncertainties, and ust∈Rn is the super-twisting robust control term, designed to suppress residual errors and external disturbances. The design of ust is as follows: 7ust=-α|s|1/2sign(s)-β∫sign(s)dt, with α=diag{α1,…,αn} and β=diag{β1,…,βn} being diagonal matrices of positive control gains.

To ensure boundedness of the neural network weights and to avoid a stringent cancellation design in the Lyapunov derivative, a hard projection operator is introduced for the weight adaptation law. Let the raw update direction matrix be Y=ΓΦ(x)sT. The adaptive law is as follows: 8W^˙=Proj(W^,Y)=Y,‖W^‖F<Wmax⁡or‖W^‖F=Wmax⁡andtr(W^TY)≤0Y-tr(W^TY)‖W^‖F2W^,‖W^‖F=Wmax⁡andtr(W^TY)>0, where Γ is a positive learning rate matrix, tr(⋅) denotes the matrix trace, and ‖⋅‖F is the Frobenius norm.

Remark 3. When the norm of the estimated weights ‖W^‖F reaches the preset safety boundary Wmax⁡ and the update direction continues to point outward, the outward normal update component is forcibly removed, limiting the update direction to the tangent of the boundary. This mathematically guarantees globally and strictly that ‖W^‖F≤Wmax⁡ holds at all times.

To intuitively illustrate the overall architecture of the proposed RBF-STSMC, the complete system block diagram is depicted in Fig. 2. As shown in the figure, the composite control law is driven by the tracking error and the sliding variable. It consists of three primary modules: the nominal dynamic compensation based on prior knowledge, the RBF neural network feedforward term for online approximation of unknown dynamics, and the super-twisting robust term designed to suppress chattering and reject residual disturbances.

Substituting the control law Eq. (6) into Eq. (5) yields the closed-loop sliding-surface dynamics: 9s˙=-W̃TΦ(x)+δ(t)+ust, where W̃=W^-W∗ is the weight estimation error, and δ(t)=M0-1(q)ε(x)+τd is a bounded composite term comprising the network approximation error and external disturbance.

We define the total lumped disturbance on the sliding surface as Δtotal(t)=-W̃TΦ(x)+δ(t). Given the projection operator (‖W^‖F≤Wmax⁡) and assumption 2, the ith component of Δtotal(t) strictly satisfies 10|Δtotal,i(t)|≤Li, where Li>0 is an existing but unknown constant bound, providing a rigorous prerequisite for robust control.

Theorem 1. We consider system (2) to satisfy assumptions 1–2. Under the sliding surface (4), control law (6)–(7), and adaptive law (8), if the gains αi and βi satisfy certain conditions then the sliding variable s converges to the origin in finite time.

Proof. We write the i subsystem dynamics in the standard super-twisting form: 11s˙i=-αi|si|1/2sign(si)+vi+Δtotal,i(t)v˙i=-βisign(si), where vi is the super-twisting integral state. From Eq. (10), the disturbance satisfies |Δtotal,i(t)|≤Li.

We construct the transformed state vector ζi=[ζi1,ζi2]T=|si|1/2sign(si),viT. Differentiating and writing in a matrix form gives 12ζ˙i=12|ζi1|Aiζi+BΔtotal,i(t), where Ai=-αi1-2βi0,B=10. We consider the strongly robust Lyapunov function, 13Vi(ζi)=ζiTPiζi, with Pi=124βi+αi2-αi-αi2, to be strictly symmetric positive definite. Differentiating Vi along the system trajectory yields the following: 14V˙i=12|ζi1|ζiTAiTPi+PiAiζi+1|ζi1|Δtotal,i(t)BTPiζi. Since |Δtotal,i(t)|≤Li, algebraic bounding arguments show that if the control gains satisfy 15αi>1.5Li,βi>1.1Li then there exists a positive definite matrixQ̃i>0 such that 16V˙i≤-12|ζi1|ζiTQ̃iζi≤-λmin⁡(Q̃i)2|ζi1|‖ζi‖2. Using Rayleigh quotient inequalities λmin⁡(Pi)‖ζi‖2≤Vi≤λmax⁡(Pi)‖ζi‖2 and the fact that |ζi1|≤‖ζi‖≤Vi/λmin⁡(Pi), we obtain 17V˙i≤-γiVi1/2, where γi=λmin⁡(Q̃i)λmax⁡1/2(Pi)λmax⁡(Pi)>0.

By the Lyapunov finite-time stability comparison lemma, Eq. (17) implies that Vi converges to zero in finite time Tr≤2Vi1/2(0)γi. Consequently, the transformed state ζi (i.e., si and vi) converges to the origin in finite time. Then, from the linear sliding surface of Eq. (4), the tracking error e converges asymptotically to zero.

Remark 4. Conventional adaptive SMC often uses the adaptive law to exactly cancel uncertainty terms, easily leading to gain drift. In contrast, the decoupled proof framework adopted here utilizes the RBF network to physically reduce the total disturbance bound L. Consequently, the required super-twisting gains (α, β) can be substantially relaxed, effectively reducing energy consumption and chattering while maintaining finite-time stability guarantees.

The UR10 6-degree-of-freedom manipulator is taken as the simulation object, and the nominal dynamic parameters are adopted from the experimental identification results reported in Petrone et al. (2025). To simulate parameter uncertainties, the actual model introduces perturbations on the basis of the nominal one: the masses of joints 2 and 3 are increased by 20 %, the centers of mass are shifted, and the inertia is increased by 5 %, while joint 6 remains at its nominal value to reflect asymmetric uncertainty. To accurately capture the nonlinear friction characteristics (e.g., Coulomb and viscous friction) while avoiding the numerical chattering and algebraic loops caused by discontinuous signum functions in simulations, a smooth sigmoidal friction model is introduced. The friction torque τf,i(q˙i) for the ith joint is formulated as τf,i(q˙i)=ci,1+ci,1q˙i+ci,31+e-ci,4(q˙i+ci,5), where q˙i is the joint velocity, and ci,1 to ci,5 are the identified friction coefficients for each joint. Based on experimental data, the coefficient vectors ci=[ci,1,ci,2,ci,3,ci,4,ci,5] for joints 1 to 6 are set as follows: joint 1 – [1.064, -1.006, 2.050, 7.946, -0.018]; joint 2 – [0.994, 0.956, -2.401, -59.953, -0.001]; joint 3 – [0.679, -0.812, 1.647, 19.825, -0.005]; joint 4 – [0.315, -0.176, 0.468, 134.898, -0.018]; joint 5 – [0.224, -0.192, 0.476, 331.442, -0.011]; and joint 6 – [0.235, -0.245, 0.598, 459.193, -0.013].

The external disturbance is defined as τd=0.5sin⁡(2πt)⋅I6 N m. The disturbance amplitude of 0.5 N m is selected to represent typical unmodeled torques and load fluctuations. As shown in Fig. 3, a fixed-gain study with disturbance amplitudes of 0.1, 0.5, and 2.0 N m reveals that the method maintains reliable tracking (RMSE < 0.005 rad) for disturbances up to at least 2.0 N m. For larger disturbances, the gains can be adjusted using Eq. (15) based on the estimated upper bound of the total disturbance Li.

The desired trajectory starts from the initial pose q0=[0,-π/2,0,-π/2,0,0]T rad and moves to the target pose qf=[0.316,0.672,0.506,0.672,0,1.255]T rad. A fifth-order polynomial trajectory is adopted, with a motion duration of T=10 s and a control period of dt=0.001 s. To prevent numerical overflow of the integral term in the super-twisting control (Eq. 7), saturation limits of ±100 N m are applied to the integral state during the discrete-time simulation using Euler integration with step size dt=0.001 s. The controller parameters are as follows: Λ=diag(30,30,30,25,25,40); super-twisting gains α=diag(10,10,10,25,25,80), β=diag(20,20,20,50,50,150); RBF network hidden layer nodes N=120; centers are obtained by K-means clustering from trajectory data; widths are set to the average distance between all centers; learning rate Γ=0.5I; and weight upper bound Wmax⁡=200. The weight upper bound Wmax⁡=200 is empirically chosen based on the estimated Frobenius norm of the ideal weight matrix. As shown in Fig. 4, a sensitivity analysis with Wmax⁡=50, 100, 200, and 400 shows that the average tracking RMSE remains 0.00054 rad, indicating that the control performance is largely insensitive to Wmax⁡ as long as it is set sufficiently large.

Figure 5 shows the time evolution of position tracking errors for each joint under three control schemes: pure RBF, SMC, and the proposed composite RBF-STSMC strategy. Under pure RBF control, the lack of a sliding-mode robust term leads to significant deterioration in tracking accuracy when facing parameter perturbations and external disturbances. As shown in Fig. 5a, the steady-state errors of joints 4, 5, and 6 are as high as 0.7108, 0.8159, and 0.4795 rad, respectively, while their maximum tracking errors reach 1.2001, 0.8865, and 1.0151 rad. This indicates that an RBF neural network alone lacks effective disturbance rejection capability without sliding-mode compensation. The SMC strategy achieves significantly better tracking accuracy due to the robust action of its fixed-gain switching term; as shown in Fig. 5b, the steady-state errors of joints 4 and 5 drop to 0.0011 and 0.0077 rad, respectively. In contrast, the proposed RBF-STSMC composite strategy exhibits superior tracking accuracy on all joints. As shown in Fig. 5c, the steady-state errors of joints 4 and 5 are further reduced to 0.0005 and 0.0017 rad, representing improvements of over 50 % and 78 % compared to pure SMC, respectively. Except for joint 5, all other joints achieve steady-state errors better than 1×10-3 rad, strongly verifying the fast convergence and robust disturbance attenuation capability of the proposed method.

Figure 6 shows the time histories of the sliding variables si for each joint under the SMC and RBF-STSMC strategies. Both methods guarantee boundedness of the sliding variables. For the proposed RBF-STSMC composite strategy, the maximum absolute values |si| of the sliding variables for joints 4, 5, and 6 are 0.1483, 0.1152, and 0.0592, respectively, whereas corresponding values for pure SMC are 0.2703, 0.9733, and 0.0639. Moreover, the composite method reduces the peak sliding variable of joint 5 by 88 %. The composite controller exhibits excellent transient response performance, with sliding-surface values being quickly constrained within a narrow boundary layer. This advantage stems from the finite-time convergence of the super-twisting control law and the RBF network's fast compensation for unknown nonlinearities. In contrast, although the pure SMC strategy eventually converges, its fixed-gain structure struggles to adapt to time-varying uncertainties, leading to larger transient deviations in the sliding surface.

As shown in Fig. 7, the maximum torque among the three methods occurs at joint 2, with values of 63.13 N m (RBF), 62.13 N m (SMC), and 62.24 N m (RBF-STSMC). All peak torques are well below the preset actuator saturation limits. However, the qualitative characteristics of the torque signals differ fundamentally. The proposed RBF-STSMC outputs a continuous and smooth control torque, free of the high-frequency chattering commonly observed in traditional SMC. It achieves significantly better tracking accuracy alongside smooth control signals, indicating strong engineering feasibility.

Figure 8 compares the evolution of the Frobenius norm of the RBF weight matrix under pure RBF and the proposed RBF-STSMC composite control configurations. Under pure RBF control, the weight norm exhibits a significant growth trend, reaching a final value of 5.72. This reflects the network's attempt to independently learn all unknown system dynamics and disturbance characteristics without the aid of a robust control term. Under the proposed RBF-STSMC strategy, the weight norm stabilizes at 0.12 after an initial 3.7 s transient growth. This low norm value is attributed to effective synergy: the super-twisting sliding-mode term undertakes the main disturbance rejection task, thereby limiting the neural network's role to fine compensation of unmodeled dynamics.

Figure 9 shows the three-dimensional Cartesian space trajectory of the end effector. This simulation is a point-to-point joint-space motion, with start and end points marked by a green circle and red triangle. The pure RBF controller yields an unacceptable steady-state Cartesian error of 0.1031 m, with a maximum deviation of 0.1444 m. The pure SMC method reduces the steady-state error to 0.0007 m, while the proposed RBF-STSMC composite method further improves it to 0.0002 m. These results fully confirm that the composite controller's joint-space error attenuation effectively transfers to the operational space, ensuring that the end effector tracks the prescribed path without noticeable overshoot or residual oscillation, ultimately achieving precise control.

To further quantify control performance, the integral of time-weighted absolute error (ITAE) is adopted as a comprehensive metric, defined as ITAE=∫0Tt|e(t)|dt. A smaller ITAE value indicates better combined performance in terms of dynamic response speed and steady-state accuracy. The ITAE values for each joint are listed in Table 1.

Compared with pure SMC, the total ITAE of the RBF-STSMC controller is reduced by 86.1 %. In particular, the reductions for joints 5 and 4 reach 87.1 % and 70.9 %, respectively, indicating that the introduction of the RBF neural network effectively reduces the burden on the super-twisting gains and improves steady-state accuracy. Compared with pure RBF control, the total ITAE is reduced by as much as 99.7 %, fully demonstrating the necessity of the sliding-mode robust term in suppressing parameter perturbations and external disturbances. These results verify the superior performance of the proposed composite control strategy in terms of tracking speed and steady-state convergence accuracy.

To further demonstrate the generality of the proposed method beyond point-to-point motions, its tracking performance was also evaluated under a continuous sinusoidal trajectory. Each joint tracks an independent sinusoidal trajectory with different frequencies and amplitudes to simulate industrial continuous operation conditions.

As shown in Fig. 10 and Table 2, the proposed RBF-STSMC controller maintains ultra-high tracking accuracy under periodic trajectories, with all joint RMSE values being less than 0.003 rad. Compared with SMC and pure RBF neural network control, the proposed method achieves significant improvements in tracking precision under continuous industrial motions. These results further verify the superior performance, generality, and robustness of the proposed composite control strategy.