Integral Nonlinear Hyperplane SMC for Robot Control

• INH-SMC is a robust nonlinear control approach for wheeled mobile robots that integrates differential-flatness linearization with terminal sliding mode and nonlinear hyperplane injection for disturbance rejection.
• It employs integral terminal sliding surfaces and a nonlinear hyperplane extension to ensure finite-time convergence, accurate trajectory tracking, and reduced control effort.
• Experimental validation on TurtleBot3 platforms demonstrates significant performance improvements in tracking accuracy and energy efficiency compared to standard sliding mode controllers.

Integral Nonlinear Hyperplane-Based Sliding Mode Control (INH-SMC) is a robust nonlinear control strategy developed for wheeled mobile robots (WMRs) to achieve accurate trajectory tracking in environments subject to significant disturbances such as strong winds or uneven surfaces. The approach integrates differential flatness-based feedback linearization, an integral terminal sliding mode manifold, and a nonlinear hyperplane (NTSMC) injection to guarantee finite-time convergence of tracking errors, disturbance rejection, and reduced control effort. Practical validation on TurtleBot3 platforms demonstrates the superior performance of INH-SMC compared to standard flatness-based sliding mode control under realistic disturbance conditions (Rehman et al., 23 Dec 2025).

1. System Model and Differential-Flatness Linearization

INH-SMC begins with a differential-drive mobile robot, where the pose is denoted

$\Omega = [\,x,\;y,\;\theta\,]^\top$

and wheel angular velocities are $(u_\ell, u_r)$. The robot’s kinematic model is: $$\dot x = v \cos\theta,\quad \dot y = v \sin\theta,\quad \dot\theta = w$$ with $v = \frac{r}{2}(u_\ell + u_r)$ (forward velocity) and $w = \frac{r}{D}(u_r - u_\ell)$ (angular velocity).

The system's differential flatness is exploited by selecting the flat outputs $\Gamma = [x, y]^\top$, allowing input-output linearization via the virtual inputs $\dot v$ and $\dot\theta$. The resulting flat-domain model, incorporating matched disturbances $(\varpi_x, \varpi_y)$ on the accelerations, is: $$\dot\Gamma_{i1} = \Gamma_{i2},\quad \dot\Gamma_{i2} = v_i + \varpi_i,\quad i\in\{x,y\}$$ This decoupling yields two disturbed double-integrator systems in the flat coordinates, setting the stage for advanced sliding mode design (Rehman et al., 23 Dec 2025).

2. Sliding Surface Design and Nonlinear Hyperplane Augmentation

The controller defines the tracking errors: $$e_i = \Gamma_{i1} - \Gamma_{id},\quad \dot e_i = \Gamma_{i2} - \dot\Gamma_{id},\quad i=x,y$$ A nonlinear integral terminal sliding mode (ITSMC) surface is specified as: $s_i(t) = \kappa_{i1}\,e_i + \kappa_{i2}\int_0^t |e_i(\tau)|^{\Phi_i} \,\sign(e_i(\tau))\,d\tau$ with $\kappa_{i1},\kappa_{i2}>0$ and $\Phi_i\in(0.5,1)$. This structure ensures "terminal" (finite-time) convergence for sliding variable dynamics.

Enhanced finite-time convergence is guaranteed through a nonlinear hyperplane extension: $\sigma_i = s_i + \mu_i\,|z_i|^{\beta_i}\,\sign(z_i); \quad z_i = \dot s_i, \; \mu_i>0, \; \beta_i > 1$ The hyperplane term enables faster reaching by manipulating the rate of convergence of $\dot s_i$ directly, critical for disturbance rejection and high-precision trajectory tracking (Rehman et al., 23 Dec 2025).

3. Control Law Synthesis

The input for each flat output channel is decomposed as: $v_i = v_{eq,i} + v_{sw,i}$ where the equivalent control

$v_{eq,i} = \frac{1}{\kappa_{i1}}\left[ -\,\kappa_{i2}\,|e_i|^{\Phi_i}\,\sign(e_i) + \ddot\Gamma_{id} \right]$

ensures trajectory following in the absence of disturbances, while the switching term

$v_{sw,i} = -\,\frac{1}{\kappa_{i1}} \left( \Upsilon_{i1}\,\sigma_i + \Upsilon_{i2}\,\sign(\sigma_i) \right)$

with $\Upsilon_{i1},\,\Upsilon_{i2}>0$, robustifies against matched disturbances $\varpi_i$ by dominating their contribution on the sliding manifold (Rehman et al., 23 Dec 2025). Saturation of control inputs is enforced to remain within actuator constraints.

4. Finite-Time Stability Analysis

A Lyapunov candidate,

$V = \tfrac{1}{2}(\sigma_x^2 + \sigma_y^2)$

is constructed for the reaching phase. Under the condition $|\kappa_{i1}\varpi_i|\le\Upsilon_{i2}$, one obtains: $$\dot V \le -\sum_{i}\mu_i\beta_i\,|\dot s_i|^{\beta_i-1}\Upsilon_{i1}\sigma_i^{2} \le 0$$ ensuring finite-time convergence to $\sigma_i = 0$. The subsequent sliding phase yields reduced dynamics for the higher-order sliding variable $z_i = \dot s_i$: $\dot z_i = -\frac{1}{\mu_i\beta_i} |z_i|^{2-\beta_i}\sign(z_i)$ which is finite-time stable for $2-\beta_i\in(0,1)$. The entire error cascade (from $\sigma_i$ to $e_i$) thus converges in finite time, with explicit settling times for each layer expressed in terms of switching and sliding manifold parameters (Rehman et al., 23 Dec 2025).

5. Parameter Selection and Practical Implementation

The main tuning parameters are:

• $\kappa_{i1},\,\kappa_{i2}>0$: increase $\kappa_{i2}$ for faster terminal convergence; large values can increase control energy.
• $\Phi_i\in(0.5,1)$: $\Phi_i$ near 0.5 yields faster convergence, approaching 1 yields smoother but slower dynamics.
• $\mu_i>0,\,\beta_i>1$: large $\mu_i$ or small $\beta_i$ accelerates initial convergence.
• $\Upsilon_{i1},\Upsilon_{i2}$: set $\Upsilon_{i2} \gtrsim \sup |\kappa_{i1}\varpi_i|$ to out-compete disturbances; $\Upsilon_{i1}$ tunes rate on the manifold.

Robust practical implementation requires:

• Chattering mitigation via boundary layer or high-frequency approximations of the $\sign(\cdot)$ function.
• Low-pass filtering of $\dot e_i$ and $\dot s_i$ to suppress sensor noise.
• Rate limiters and command saturation to respect physical constraints on velocity and acceleration.
• Real-time control loop execution at frequencies above the SMC switching bandwidth.
• Safety measures such as ROS node-enforced command saturation for actuator protection (Rehman et al., 23 Dec 2025).

6. Experimental Validation and Quantitative Performance

The approach was experimentally validated on a TurtleBot3 platform subjected to simulated "rough floor" and "wind gust" disturbances using foam panels and a blower. INH-SMC was systematically compared to flatness-based SMC (FBSMC) during extreme transient disturbances at $t \approx 70$ s.

Quantitative performance is summarized below:

Controller	IAEₓ	IAEᵧ	ISEₓ	ISEᵧ	$P_{\text{avg}}$
FBSMC	7.0131	6.9040	0.9719	0.5829	0.0967
Proposed	3.8904	6.4075	0.2465	0.3763	0.0293

Here, IAE is the integral of absolute error, ISE the integral of squared error, and $P_{\text{avg}}$ the mean control effort. INH-SMC reduces IAEₓ by approximately 50% and mean power by a factor of ~3, while maintaining velocities within hardware limits (0.22 m/s, 2.84 rad/s) (Rehman et al., 23 Dec 2025).

7. Design Implications and Scope

INH-SMC merges the disturbance-rejection capabilities of nonlinear sliding mode architectures with the flatness-enabled decoupling of mobile robot kinematics. The architecture is directly applicable to differential-drive WMRs featuring flat kinematics and may, in principle, generalize to other flat-underactuated systems subject to matched uncertainties. Experimental verification on realistic hardware, combined with Lyapunov-based finite-time convergence proofs, establishes the method's viability and substantiates its advantages in disturbed mobile robot tracking scenarios (Rehman et al., 23 Dec 2025).