Source-linked AI summary
Derivative-Free Generalized Multivariable Super-Twisting Control for Constrained Euler-Lagrange Systems
Chidre Shravista Kashyap, Jishnu Keshavan
TL;DR
Robotic manipulators must track trajectories through abrupt payload-load changes despite limited actuation and uncertain plant knowledge, while existing super-twisting designs rely on a disturbance-derivative condition that non-smooth EL disturbances violate. The paper develops a derivative-free generalized multivariable controller with adaptive and feedforward perturbation handling, saturation treatment, and a time-shifting barrier. Simulation and FR3 experiments are reported to confirm the scheme’s properties under actuator saturation and non-smooth disturbances.
Problem
Existing super-twisting designs rely on a bounded disturbance derivative, an assumption violated by the non-smooth disturbances and discontinuous first-order sliding-mode dynamics of EL systems.
Source: §1 · p. 1Method
A continuous fractional-power integral term removes derivative dependence, while adaptation and feedforward compensation handle state-dependent perturbations and a time-shifting barrier restores the prescribed envelope after escape.
Source: §1 · p. 2, §5 · pp. 13–14Results
Simulation and experimental studies on the FR3 manipulator confirm the proposed properties under actuator saturation and non-smooth disturbances.
Source: §4 · p. 9, §5 · p. 14Takeaways & Limitations
The design requires only mass-matrix norm bounds and converges to a user-specified bound within a user-prescribed time, independently of initial conditions and unknown system parameters.
Source: §1 · p. 2, §5 · p. 14With limited control authority, prescribed-time convergence may not be achievable from an arbitrarily large initial offset; longer tc expands the feasible domain, while lowering η reduces u_min at the cost of a smaller domain.
Source: §3.1 · pp. 4–5
Abstract
from arXiv · showhide
Robotic manipulators performing payload lift-and-transfer must track prescribed trajectories under abrupt load changes, with limited actuation and inaccurate plant knowledge. Such plants are inherently described by multivariable Euler--Lagrange~(EL) dynamics with cross-coupling established through inertial and Coriolis terms. The super-twisting algorithm~(STA) is the standard approach to regulating a first-order sliding mode~(FOSM) in such settings, yet existing designs are largely restricted to scalar sliding variables, and rest on a disturbance-derivative assumption that EL systems violate. In particular, the FOSM dynamics depend on the disturbance and become discontinuous under the step changes in load produced by payload engagement and release, rendering that derivative ill-defined. This study resolves both with a derivative-free generalized multivariable super-twisting framework. The discontinuous sign-function integral term is replaced with a continuous fractional-power term whose dynamics are independent of the disturbance derivative, while the known state-dependent scaling of the perturbation is incorporated into both the control law and the adaptive gain update. A time-shifting barrier function guarantees prescribed-time convergence to a user-specified bound independently of initial conditions, with the design relying only on mass-matrix norm bounds. Simulation and experimental results on a 7-DoF manipulator under actuator saturation are used to validate the proposed scheme.
1 INTRODUCTION
The introduction identifies a mismatch between standard super-twisting designs and multivariable Euler–Lagrange systems under non-smooth disturbances. It proposes a derivative-free, saturation-aware multivariable framework with prescribed-time convergence.
1 INTRODUCTION
- Existing STA designs retain a bounded disturbance-derivative requirement that non-smooth load changes violate, leaving their stability guarantees uncertifiable for EL systems under general bounded disturbances.The comparison organizes designs by derivative-free operation, prescribed bound and time, and feasibility.Source: §1 · pp. 1–2, Table 1 · p. 2
- A continuous fractional-power integral term removes disturbance-derivative dependence, while a Kronecker-product Lyapunov structure and matrix saturation coefficient address multivariable EL coupling.The proposed gain adaptation handles unknown perturbation magnitude, and feedforward compensation handles its known state-dependent factor; only mass-matrix norm bounds are required.Source: §1 · p. 2
- A time-shifting barrier resets the convergence envelope after escape, guaranteeing convergence to ε within tc independently of initial conditions and unknown system parameters.The adaptive gain remains bounded, and the steady-state bound scales sublinearly rather than linearly with the relevant design parameter.Source: §1 · p. 3
- The saturation-consistent design uses a nonlinear equality transformation and filtered saturation coefficient to bound control action and mitigate integral windup without switching.The introduction also states that its admissible gain conditions are more relaxed than those of.Source: §1 · p. 3
2 Preliminaries
The preliminaries define notation and convergence concepts, then formulate a saturated multivariable Euler–Lagrange tracking problem with bounded disturbance and limited control authority.
2.1 Notations, Definitions and Lemmas
- The multivariable sign function scales a unit vector by a power of its norm, and practical prescribed-time convergence means remaining within ε for all t ≥ Tc.Solutions to the stated possibly discontinuous system are interpreted in the Filippov sense.Source: §2.1 · p. 3
- For symmetric positive definite A and B, their Kronecker product is positive definite and its minimum eigenvalue is the product of their minimum eigenvalues..Source: §2.1 · p. 3
2.2 Problem Statement
- The plant model includes mass, Coriolis, gravity, friction, external disturbance, and a componentwise saturated control input.The saturation bound is known, and the system is assumed input-to-state stable.Source: §2.2 · p. 3
- The mass matrix and its inverse are symmetric positive definite; only mass-matrix norm bounds are known, while the Coriolis, gravity, and friction terms and their bounds are unknown.The stated norm properties bound Coriolis, gravity, and friction contributions.Source: §2.2 · p. 4
- With bounded disturbance and reference velocity and acceleration, the controller must drive s = ė + Γe into a predefined bound within a predefined time under limited actuation.The objective allows a state-dependent perturbation and does not require its magnitude to be known.Source: §2.2 · p. 4
3 Main Result
The section establishes prescribed-time convergence for a constrained multivariable super-twisting controller, subject to a feasibility condition on control authority. Its design combines an adaptive controller with a time-shifting barrier that restores convergence after the prescribed envelope is exceeded.
3.1 Feasibility Analysis
- Under input constraints, exact sliding convergence within tc is guaranteed only when the feasibility condition holds; the set D identifies initial conditions from which trajectories reach the origin in that time.The design guidance states that increasing tc expands D, whereas decreasing η reduces required control authority at the cost of a smaller D.Source: §3.1 · pp. 4–5
3.2 Adaptive Multivariable Super-Twisting Control Design
- A nonlinear equality transformation converts the constrained input into an unconstrained variable, while a filtered saturation coefficient represents input saturation in controller synthesis.The coefficient decreases as saturation becomes more extensive; filtering keeps it available for feedback while preserving positive bounds.Source: §3.2 · pp. 5–6
- The time-shifting barrier resets the convergence envelope after escape, keeping the barrier well-defined and enabling recovery for bounded disturbances.The envelope decreases from an unbounded initial value to ε by tc; earlier barrier approaches could lose stability after sudden disturbance changes.Source: §3.2 · p. 6
3.3 Stability analysis
- With sufficient control authority satisfying the task-feasibility condition, the controller drives the sliding variable below ε within prescribed time tc, independently of initial conditions and unknown system parameters.If the envelope is escaped, it is reset and convergence to ε is guaranteed within tc thereafter.Source: §3.3 · pp. 6, 9
- The steady-state ultimate bound can be selected independently of initial conditions and tc, allowing tc to expand the attraction domain without changing achievable steady-state accuracy.For sufficiently large gains, the bound is approximately √(με).Source: §3.3 · p. 9
- A dead-zone adaptation prevents the gain from increasing while the sliding variable is within ε or the input is saturated; envelope escape in saturation is followed by re-convergence within tc.When unsaturated and outside ε, the adaptation matches the original update.Source: §3.3 · p. 9
4 Results and Discussion
FR3 experiments test nominal and perturbed trajectory tracking, while numerical and experimental comparisons assess the proposed controller against prior designs.
4.1 Experimental Validation
- With no disturbance, sliding error converged at tc=6 s under saturation; convergence to ε persisted for 10°–30° offsets and tc=3–6 s.After the t=13 s reference reset, the sliding variable re-converged and steady-state accuracy was preserved.Source: §4.1 · p. 10
- Experimental total variation per input channel fell monotonically over α=0.6–0.8, rose slightly at 0.9 from encoder-noise amplification, while tracking RMSE changed only on the order of 10^-2.The noise-free simulation recovered monotonicity across the full α range.Source: §4.1 · p. 11
- Smooth 0.5- and 1-kg payloads preserved comparable errors without envelope violation; impulsive cable loading triggered barrier resets and re-convergence despite saturation.The dead-zone arrested gain adaptation during saturated intervals, and steady-state metrics were recovered after both the taut-cable and manual perturbation events.Source: §4.1 · pp. 11–12
4.2.1 Numerical Comparisons
- Steady-state errors are on the order of 10^-3 for proposed scheme (17), versus 10^-1 for and 10^2 for, in joint position and velocity.The comparison is corroborated by the FR3 tracking responses in Figure 5.Source: §4.2.1 · p. 12, Table 3 · p. 13, Figure 5 · p. 13
4.2.2 Experimental comparison
- Against, experiments show substantially lower steady-state error; and diverged under nominal conditions, while encoder noise widened the margin.The and trials were terminated when instability began, precluding meaningful experimental comparisons under payload perturbation.Source: Table 3 · p. 13, §4.2.2 · p. 13
5 Conclusion
The proposed framework combines derivative-free multivariable super-twisting control with adaptive and feedforward perturbation treatment and saturation handling. It provides user-prescribed-time convergence to a user-specified bound and is evaluated on the FR3 manipulator under saturation and non-smooth disturbances.
5 Conclusion
- A continuous fractional-power term replaces the discontinuous sign-function integral term, removing structural dependence on the disturbance derivative.Joint gain adaptation and direct feedforward compensation address state-dependent perturbation scaling without requiring knowledge of perturbation magnitude.Source: §5 · p. 13
- The framework converges in a user-prescribed time to a user-specified bound, independently of initial conditions and unknown system parameters.Simulation and experimental studies on the FR3 manipulator confirm these properties under actuator saturation and non-smooth disturbances.Source: §5 · p. 14
- A nonlinear equality transformation with a filtered saturation coefficient accommodates input saturation.A time-shifting barrier restores the convergence guarantee whenever the trajectory escapes the prescribed envelope.Source: §5 · p. 14