Velocity-Direction-Based Adaptive Super-Twisting Control of a Nonlinear Car-Like Mobile Robot under Steering-Rate Perturbations

The numerical evidence consists of five paired scenarios, S0–S4, using the same controller functions and tuning throughout. S0 retains the original comparison, while S1–S4 vary curvature, speed, disturbance amplitude, and initial error individually. Section 4.2 also retains an analytical heading-based super-twisting reference formulation to explain the different input–output pathways; that formulation is not the adaptive heading-feedback baseline used to generate S0–S4. Sections 4.4 and 6.1 identify the scope of each formulation so that the numerical comparison is attributed to the controllers actually evaluated.

The main contributions of this work are summarized as follows:

•A center-based nonlinear kinematic formulation explains why the body heading $\psi$ and effective velocity direction $\theta_v=\psi+\beta$ have different roles during cornering.

•The control design separates nominal heading dynamics from the geometric steering contribution to the center-velocity-direction channel.

•A velocity-direction-based ASTSMC formulation regulates the angular variable governing center translation through a direct steering-rate command.

•Ideal-regime boundedness and sliding-convergence arguments are stated with their assumptions and separated from the finite-horizon evidence under sampling and saturation.

•Five paired scenarios evaluate tracking, post-disturbance recovery, steering input, and adaptive gains with unchanged tuning, including the stronger-disturbance case in which the baseline loses accurate tracking.

Section 2 presents the center-based model and its physical interpretation. Section 3 defines the tracking objectives and analytical disturbance assumptions. Section 4 presents the controller formulations and distinguishes them from the implemented comparison. Section 5 discusses conditional stability and practical limitations. Section 6 reports the S0–S4 settings and results, followed by engineering interpretation in Section 7 and conclusions in Section 8.

3.1 Control objective

The objective of this study is to design the control inputs

$\mathrm{u}(t)=\left[a(t) \quad \omega_\delta(t)\right]^T$        (9)

such that center-position and direction errors are reduced while the applied steering commands respect their prescribed limits. The design objective is accurate tracking and recovery in the presence of bounded perturbations. Conditional ideal-regime analysis and the finite-horizon constrained simulations provide different types of evidence and are interpreted separately.

3.2 Reference trajectory and tracking errors

Let the desired trajectory be described by the reference pose

$\mathrm{q}_d=\left[\begin{array}{lll}x_d & y_d & \psi_d\end{array}\right]^T$       (10)

The trajectory-tracking errors are defined within the robot body-fixed frame as

$\left[\begin{array}{l}e_x \\ e_y \\ e_\psi\end{array}\right]=\left[\begin{array}{ccc}\cos \psi & \sin \psi & 0 \\ -\sin \psi & \cos \psi & 0 \\ 0 & 0 & 1\end{array}\right]\left[\begin{array}{l}x_d-x \\ y_d-y \\ \psi_d-\psi\end{array}\right]$             (11)

Here, $e_x$ and $e_y$ denote the longitudinal and lateral position errors expressed in the body-fixed frame, while $e_\psi$ denotes the heading-angle error. Through this body-frame representation, the position error is decomposed into along-track and crosstrack components relative to the robot orientation. In particular, the lateral error $e_y$ is utilized within the steering-control design to quantify the deviation of the geometric center from the desired path.

3.3 Geometric slip angle and effective motion direction

Within the adopted center-based kinematic bicycle model, the translational motion of the geometric center is governed by Eqs. (3)–(4), wherein the steering-induced geometric slip angle $\beta$ is defined by Eq. (2).

In this formulation, $\beta$ characterizes the geometric offset between the body heading angle and the true velocity direction of the geometric center. Consequently, the effective velocity-direction angle is established in Eq. (1).

These relationships demonstrate that the translational trajectory of the geometric center is dictated by the effective velocity-direction angle $\theta_\nu$, rather than by the body heading angle $\psi$ alone. Consequently, $\theta_v$ is proposed as the primary steering variable for the advanced tracking controller. By adopting this formulation, the steering-induced geometric offset $\beta$ is directly incorporated into the regulated motion direction, a feature that is particularly critical during cornering maneuvers.

3.4 Compared steering variables

The analytical discussion compares heading and velocity-direction steering objectives within the same nominal kinematic geometry. In the heading formulation, the angular error is $e_\psi$ in Eq. (11). For the proposed formulation, $\theta_{v, d}$ denotes the desired center-velocity direction associated with the guidance law. Its error is

$e_{\theta_v}=\theta_{v, d}-\theta_v$          (12)

The heading objective concerns body orientation, whereas the proposed objective concerns the center-velocity direction. The desired tracking behavior is small planar error together with the relevant angular error, summarized as

$e_x(t) \rightarrow 0, \quad e_y(t) \rightarrow 0$         (13)

together with either

$e_\psi(t) \rightarrow 0 \quad$ or $\quad e_{\theta_v}(t) \rightarrow 0$          (14)

for the selected formulation. These expressions describe tracking objectives, not an unconditional convergence claim for the saturated numerical implementation.

The numerical comparison evaluates the proposed velocity-direction controller against the implemented adaptive heading-feedback baseline under identical scenario conditions. It does not isolate output choice from controller structure or tuning.

3.5 Disturbance and actuator constraints

For the analytical steering-actuator model, a matched perturbation $d_m(t)$ is introduced into the steering-rate dynamics. On intervals where the regularity assumptions apply, it satisfies

$\left|d_m(t)\right| \leq \bar{d}_m, \quad\left|\dot{d}_m(t)\right| \leq \bar{\rho}_m$          (15)

where $\bar{d}_m>0$ and $\bar{\rho}_m>0$ represent unknown positive constants. Consequently, the perturbed steering dynamics are formulated as

$\dot{\delta}=\omega_\delta+d_m(t)$             (16)

Eq. (16) is the analytical steering-actuator disturbance model used in the derivation of Section 4. It is not the disturbance-injection rule used to generate the S0–S4 benchmark.

The numerical benchmark applies $d_\theta(t)$ directly to the effective motion-direction derivative, as specified in Eqs. (94)–(95), while the steering angle is advanced from the commanded rate and constrained to its admissible interval. The two channel descriptions are not assumed equivalent. The S0–S4 comparisons use the same implemented channel for both controllers, with its parameters held fixed except for the amplitude change in S3.

The steering angle is strictly bounded by the physical actuator constraint

$|\delta(t)| \leq \delta_{\max}$        (17)

while the steering-rate input is constrained such that

$\left|\omega_\delta(t)\right| \leq \omega_{\max}$          (18)

The steering constraints are imposed independently of the desired tracking accuracy. In the ideal analysis, boundedness of the quantities

$v \left( t \right), |\omega_{\delta } \left( t \right), |\beta \left( t \right)$     (19)

is considered under the stated assumptions. In the numerical benchmark, rate clipping and steering-angle projection enforce their respective limits; these mechanisms do not by themselves guarantee convergence of the position errors.

The following analysis concerns the ideal analytical formulations of Section 4. Conditional gain boundedness and ideal sliding convergence are discussed under the assumptions below. The heading formulation additionally requires a sufficiently regular inner-loop residual. Section 5.5 explains its relationship to sampled, constrained operation.

Assumption 1. The reference trajectory is assumed to be sufficiently smooth, such that the signals $x_{d}$, $y_{d}$, $\dot{x}_{d}$, $\dot{y}_{d}$, $\ddot{x}_{d}$, and $\ddot{y}_{d}$ are uniformly bounded.

Assumption 2. The guidance signals generated via Eqs. (20)-(24) are assumed to satisfy

$\left| v_{c} \left( t \right) \right| \leq \overline{v}_{c}, \left| \dot{v}_{c} \left( t \right) \right| \leq \overline{a}_{c}$                 (62)

$\left| \theta_{v,d} \left( t \right) \right| \leq \overline{\theta}_{d}, \left| \dot{\theta}_{v,d} \left( t \right) \right| \leq \overline{\omega}_{d}$                (63)

for specified positive constants $\overline{v}_{c}$, $\overline{a}_{c}$, $\overline{\theta}_{d}$, and $\overline{\omega}_{d}$.

Assumption 3. The steering actuator is assumed to operate strictly within the admissible region specified by Eq. (17), subject to $\delta_{\max} < \pi/2$. Concurrently, the steering-rate input is bounded via saturation, as detailed in Eqs. (42) and (60).

Assumption 4. The steering-rate perturbation $d_{m} \left( t \right)$ introduced in Eq. (16) is assumed to be bounded and sufficiently regular, satisfying

$\left| d_{m} \left( t \right) \right| \leq \overline{d}_{m}, \left| \dot{d}_{m} \left( t \right) \right| \leq \overline{\rho}_{m}$                 (64)

for defined positive constants $\overline{d}_{m}$ and $\overline{\rho}_{m}$.

Assumption 5. For the analytical heading-based formulation, the inner steering loop is assumed sufficiently responsive on the interval considered. Its steering tracking error is

$\varepsilon_{\delta} = \delta_{\psi,c} - \delta$          (65)

and the realized heading rate is written as

$r_{\psi} = r_{\psi,c} + \Delta_{\psi} \left( t \right)$         (66)

where $\Delta_{\psi}(t)$ is additionally assumed to be bounded with a bounded derivative. This residual regularity is an assumption, not a consequence of a positive inner-loop gain alone:

$\left| \Delta_{\psi} \left( t \right) \right| \leq \overline{\Delta}_{\psi}, \left| \dot{\Delta}_{\psi} \left( t \right) \right| \leq \overline{\nu}_{\psi}$             (67)

Under Assumption 3, the geometric slip angle is guaranteed to remain bounded by virtue of Eq. (2). Specifically, it satisfies

$\left| \beta \left( t \right) \right| \leq \beta_{\max} = \tan^{-1} \left( \frac{l_{r}}{L} \tan \delta_{\max} \right)$               (68)

Furthermore, given that $\left| \delta \right| < \pi/2$, the gain $G \left( \delta \right)$ defined in Eq. (45) remains strictly positive and bounded across the admissible steering domain. Consequently, bounding constants $G_{\min} > 0$ and $G_{\max} > 0$ exist such that

$0 < G_{\min} \leq G \left( \delta \right) \leq G_{\max}$            (69)

Given the boundedness of both $G \left( \delta \right)$ and $d_{m} \left( t \right)$, the induced perturbation

$\Delta_{m} \left( t \right) = G \left( \delta \right) d_{m} \left( t \right)$            (70)

is inherently bounded. Furthermore, predicated on the boundedness of $\delta$ and $\omega_{\delta}$, alongside the regularity of $d_{m} \left( t \right)$, its time derivative is similarly bounded. Therefore, positive constants $\overline{\Delta}_{m} > 0$ and $\overline{\nu}_{m} > 0$ exist such that

$\left| \Delta_{m} \left( t \right) \right| \leq \overline{\Delta}_{m}, \left| \dot{\Delta}_{m} \left( t \right) \right| \leq \overline{\nu}_{m}$                (71)

5.1 Boundedness of the common longitudinal channel

Without longitudinal clipping or projection, Eqs. (25)–(26) yield the ideal speed-tracking error dynamics

$\dot{e}_{v} = \dot{v} - \dot{v}_{c} = - k_{v} e_{v}$           (72)

Consider the Lyapunov function candidate

$V_{v} = \frac{1}{2} e_{v}^{2}$         (73)

Its time derivative is

$\dot{V}_{v} = e_{v} \dot{e}_{v} = - k_{v} e_{v}^{2} \leq 0$     (74)

Therefore, the ideal speed error decays exponentially, and $v$ remains bounded when $v_{c}$ is bounded. This exact error equation is not asserted during numerical speed or acceleration limiting.

5.2 Boundedness of the adaptive gains

From Eqs. (37)–(39), the heading-based adaptive states remain nonnegative:

$\rho_{1}^{\psi} \left( t \right) \geq 0, \rho_{2}^{\psi} \left( t \right) \geq 0, \forall t \geq 0$              (75)

Hence, by Eqs. (35) and (36),

$k_{1}^{\psi} \left( t \right) \geq k_{1,\min}^{\psi} > 0, k_{2}^{\psi} \left( t \right) \geq k_{2,\min}^{\psi} > 0$              (76)

Similarly, from Eqs. (57)–(59),

$\rho_{1}^{\theta} \left( t \right) \geq 0, \rho_{2}^{\theta} \left( t \right) \geq 0, \forall t \geq 0$             (77)

and therefore,

$k_{1}^{\theta} \left( t \right) \geq k_{1,\min}^{\theta} > 0, k_{2}^{\theta} \left( t \right) \geq k_{2,\min}^{\theta} > 0$          (78)

Moreover, for bounded sliding variables, comparison arguments applied to the adaptation laws yield

$\rho_{1}^{\psi} \left( t \right) \leq e^{-\sigma_{1}^{\psi}t} \rho_{1}^{\psi} \left( 0 \right) + \frac{\gamma_{1}^{\psi}}{\sigma_{1}^{\psi}} \sup_{0\leq \tau \leq t} \left| s_{\psi} \left( \tau \right) \right|$        (79)

$\rho_{2}^{\psi} \left( t \right) \leq e^{-\sigma_{2}^{\psi}t} \rho_{2}^{\psi} \left( 0 \right) + \frac{\gamma_{2}^{\psi}}{\sigma_{2}^{\psi}} \sup_{0\leq \tau \leq t} \left| s_{\psi} \left( \tau \right) \right|$        (80)

$\rho_{1}^{\theta} \left( t \right) \leq e^{-\sigma_{1}^{\theta}t} \rho_{1}^{\theta} \left( 0 \right) + \frac{\gamma_{1}^{\theta}}{\sigma_{1}^{\theta}} \sup_{0\leq \tau \leq t} \left| s_{\theta_{v}} \left( \tau \right) \right|$              (81)

$\rho_{2}^{\theta} \left( t \right) \leq e^{-\sigma_{2}^{\theta}t} \rho_{2}^{\theta} \left( 0 \right) + \frac{\gamma_{2}^{\theta}}{\sigma_{2}^{\theta}} \sup_{0\leq \tau \leq t} \left| s_{\theta_{v}} \left( \tau \right) \right|$              (82)

Thus positivity follows from the nonnegative initial adaptive states and the stated laws. Gain boundedness follows when the corresponding sliding variable is bounded; these estimates are not an independent global closed-loop stability proof.

5.3 Analytical heading-based formulation

Using Eq. (32), Eq. (33), and Assumption 5, the actual heading-based sliding dynamics become

$\dot{s}_{\psi} = - k_{1}^{\psi} \left( t \right) \left| s_{\psi} \right|^{1/2} \operatorname{sgn} \left( s_{\psi} \right) + z_{\psi} - \Delta_{\psi} \left( t \right)$               (83)

with the auxiliary dynamics given in Eq. (34).

The ideal heading sliding channel has a super-twisting structure with residual $\Delta_{\psi}(t)$. Finite-time convergence requires the standard sufficient-gain and perturbation-regularity conditions to hold, in addition to Assumption 5 [24, 27-29]. Under those conditions, the ideal reaching statement is

$s_{\psi} \left( t \right) = 0, \forall t \geq T_{\psi}$          (84)

Upon reaching the sliding manifold, the tracking errors satisfy

$e_{\psi} = - \lambda_{y} e_{y}$         (85)

For the analytical reference, the applied steering-rate bound follows from Eq. (42), whereas steering-angle admissibility remains part of Assumption 3. The bound on $\beta$ then follows from Eq. (2).

5.4 Velocity-direction-based formulation

Utilizing Eqs. (52)–(54), the ideal unsaturated sliding dynamics of the proposed controller are formulated as

$\dot{s}_{\theta_{v}} = - k_{1}^{\theta} \left( t \right) \left| s_{\theta_{v}} \right|^{1/2} \operatorname{sgn} \left( s_{\theta_{v}} \right) + z_{\theta} - \Delta_{m} \left( t \right)$                 (86)

subject to the auxiliary dynamics provided in Eq. (54).

The ideal direct channel has the super-twisting form with induced perturbation $\Delta_{m}(t)$. Subject to Assumption 4 and sufficient-gain conditions for the chosen super-twisting law [24, 27-29], the associated ideal finite-time reaching statement is

$s_{\theta_{v}} \left( t \right) = 0, \forall t \geq T_{\theta}$           (87)

Upon reaching the sliding manifold, the tracking errors satisfy

$e_{\theta_{v}} = - \lambda_{y} e_{y}$          (88)

The following local lateral-error interpretation assumes positive forward speed, small errors, and the stabilizing lateral-error sign convention used in the reduced analysis. It is not a complete zero-dynamics derivation for the guided, sampled benchmark. Under those local assumptions, the interpretation is expressed as

$\dot{e}_{y} = - v \sin \left( \lambda_{y} e_{y} \right) + O \left( e_{x} e_{y} \right)$            (89)

For a sufficiently small lateral error, the approximation $\sin \left( \lambda_{y} e_{y} \right) \approx \lambda_{y} e_{y}$ holds, resulting in

$\dot{e}_{y} \approx - v \lambda_{y} e_{y}$            (90)

The reduced local relation in Eq. (90) has a decaying lateral-error mode for positive $v$ and $\lambda_{y}$. It must not be interpreted as a proof of global position convergence for the complete controller, guidance dynamics, perturbation channel, and saturation mechanisms used in S0–S4.

The rate bound is enforced by Eq. (60), while steering-angle admissibility is required by Assumption 3 in the analytical model. A bounded geometric slip angle follows on that admissible interval.

Theorem 1 (conditional ideal heading reference). Consider the analytical controller of Section 4.2 under Assumptions 1–5, bounded guidance, bounded sliding variables for the gain estimates, and the sufficient super-twisting gain conditions invoked in Section 5.3. Then the ideal longitudinal variable, adaptive gains, applied steering-rate command, admissible steering angle, and geometric slip angle have the stated bounds. Where saturation is inactive and the residual regularity and sufficient-gain conditions hold, the analytical sliding variable reaches zero in finite time.

Proof. The ideal speed-error relation in Eqs. (72)–(74) gives bounded speed for bounded guidance. Eqs. (75)–(80) give positive gains and conditional gain bounds. The applied rate is limited by Eq. (42), steering-angle admissibility is assumed, and Eq. (2) bounds the geometric offset. The finite-time part follows conditionally from the super-twisting channel in Eqs. (83) and (34) when the stated sufficient conditions hold. The sufficient conditions are assumed for this ideal statement, not certified by selecting positive numerical gains alone.

Theorem 2 (conditional ideal velocity-direction formulation). For the analytical controller in Section 4.3, assume Assumptions 1–4, bounded guidance, bounded sliding variables for the gain estimates, and sufficient super-twisting gain conditions for the induced perturbation. The quantities considered are

$v \left( t \right),|\delta \left( t \right), |\omega_{\delta} \left( t \right), |\beta \left( t \right)$       (91)

The gain estimates in Eqs. (77)–(82) are conditional on bounded sliding variables, and exponential speed-error decay concerns the unlimited longitudinal law. With inactive saturation and the sufficient-gain and regularity conditions satisfied, the ideal velocity-direction sliding channel has finite-time reaching. Eqs. (89)–(90) supply only the stated reduced local interpretation, not global trajectory convergence.

Proof. Bounded ideal speed follows from Eqs. (72)–(74); positivity and conditional gain bounds follow from Eqs. (77)–(82). Eq. (60) bounds the applied rate, while Assumption 3 and Eq. (2) bound the steering angle and geometric offset. The ideal super-twisting channel in Eqs. (86) and (54) admits the conditional reaching interpretation under the assumed perturbation regularity and sufficient gains. The sampled dead-zone implementation and the different numerical perturbation channel are not covered by transferring this argument without further analysis.

5.5 Interpretation for the implemented simulations

The S0–S4 results are finite-horizon observations of the implemented benchmark, not a numerical proof of the preceding ideal statements. Rate saturation, steering-angle projection, speed and acceleration clipping, numerical differentiation, and dead-zone adaptation alter the ideal error dynamics. In particular, the analytical heading reference in Theorem 1 is not the simulated adaptive heading-feedback baseline. The plotted gain values are not certificates of sufficient super-twisting gains.

A slow heading steering loop can increase lag between requested and realized steering, enlarge the analytical residual, and delay recovery. Once rate or angle limiting prevents the requested action, the ideal reaching argument no longer guarantees a small tracking error. This is a mechanism-based interpretation, not a claim that inner-loop bandwidth was varied in S0–S4. The baseline recovery loss in S3 is reported as an observed outcome under the retained tuning.

The S0–S4 results show a consistent separation between the implemented controllers across changes in curvature, speed, disturbance amplitude, and initial error. The proposed velocity-direction controller yields smaller tracking errors and lower integrated steering-rate effort in each pair, without case-specific retuning. This is the main numerical finding. The interpretation concerns the adaptive heading-feedback comparator and motion-direction perturbation specified in Section 6, rather than the unsimulated analytical heading reference.

The disturbance and recovery intervals play different roles. Several baseline responses develop their largest position deviations after the disturbance is switched off at 24 s, demonstrating that recovery cannot be assessed from the active-window error alone. In S3, the baseline position error is dominated by longitudinal deviation; a lateral-error metric alone would hide the severity of the tracking loss. Planar error, terminal error, and steering histories should therefore be interpreted together. The direction-error jumps in S3 also reflect angular wrapping and should not be read as instantaneous physical rotations.

The center-based kinematic interpretation is unchanged: the velocity vector points along $\theta_{v} = \psi + \beta$. This provides the motivation for regulating velocity direction. It does not by itself prove that every heading-based design must perform worse, because reference compensation, feedback realization, and tuning also affect the response. For the nominal geometric relations, the center velocity is

$v = v \left[ \begin{array}{c}\cos \theta_{v} \\ \sin \theta_{v}\end{array} \right]$           (102)

Differentiation gives

$\dot{v} = \dot{v} \left[ \begin{array}{c}\cos \theta_{v} \\ \sin \theta_{v}\end{array} \right] + v \dot{\theta}_{v} \left[ \begin{array}{c}-\sin \theta_{v} \\ \cos \theta_{v}\end{array} \right]$                   (103)

Its normal acceleration and the associated qualitative inertial-force interpretation are

$a_{\perp} = v \dot{\theta}_{v}$                  (104)

$F_{\perp} = \mathrm{mv} \dot{\theta}_{v}$                    (105)

For locally circular motion,

$F_{\perp} = m \frac{v^{2}}{R}$              (106)

Here $m$ is robot mass and $R$ is the instantaneous center-path radius. These are kinematic relations, not an actuator-power or dynamic tire model. The reported $J_{\omega}$ therefore measures steering-command activity only; electrical energy, motor torque, and mechanical loading are not established by these simulations.

For low-speed automated transport or service robots, the engineering value of these tests is the observed reduction in center-path deviation and improved recovery with the selected steering authority. A system designer must assess the tracking tolerance together with the rate command, steering angle, and saturation duration, because a bounded input can coexist with unacceptable position error. S2 shows that increased speed can produce a localized transient even when aggregate tracking remains better; S3 shows that unchanged limits can coincide with severe baseline recovery failure.

At system level, trajectory generation should be coordinated with steering-angle and steering-rate capability rather than assuming that larger gains can compensate for infeasible commands. The implemented controller also requires center-position, velocity direction or reconstructed heading, steering angle, and a lateral-error derivative. A physical realization would need a consistent estimator and a measured steering actuator, and would have to account for differentiation noise and servo delay. The current plots characterize commanded motion, not motor current, torque, heating, or electrical energy. The very small simulated terminal errors are therefore numerical tracking results, not physical positioning specifications.

The evidence remains limited to one parameterized cornering-path family, deterministic sinusoidal motion-direction perturbations, exact-state feedback, fixed geometry, and fixed actuator settings. No physical robot or Robot Operating System/Gazebo validation is reported. Tire dynamics, measurement noise, delay, and input-limit sensitivity are not evaluated. The numerical findings therefore support the stated finite set of operating conditions, while the ideal analytical reference retains its separate assumptions and validation scope.