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Theory and Application on Adaptive-Robust Control of Euler-Lagrange Systems with Linearly Parametrizable Uncertainty Bound

Spandan Roy, Sayan Basu Roy, Indra Narayan Kar

arXiv:1708.01442v1eess.SYcs.SY

TL;DR

Uncertain EL control approaches can depend on constant bounds for overall uncertainty, which restrict system states a priori, and switching gains can be over- or underestimated. The paper proposes ASRC using a linear-in-parameters uncertainty bound, and reports better tracking accuracy than ASMC in the wheeled-robot experiment.

  • Problem

    Existing ASMC relies on an unknown constant uncertainty bound, an assumption described as restrictive for EL systems and insufficient for necessary robustness.

    Source: B. Experimental Results and Comparison · p. 8

  • Method

    ASRC adapts its switching gains and parameter estimates using the uncertainty-bound structure without presuming a constant upper bound on overall uncertainty.

    Source: B. Adaptive Switching-gain based… · p. 3, §IV · p. 10

  • Results

    ASRC provides better tracking accuracy than ASMC in the wheeled mobile robot experiment.

    Source: B. Experimental Results and Comparison · p. 8

  • Takeaways & Limitations

    The proposed law is reported to negotiate uncertainties that are linear or nonlinear in parameters while alleviating switching-gain overestimation and underestimation.

    Source: §IV · p. 10

Abstract

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This work proposes a new adaptive-robust control (ARC) architecture for a class of uncertain Euler-Lagrange (EL) systems where the upper bound of the uncertainty satisfies linear in parameters (LIP) structure. Conventional ARC strategies either require structural knowledge of the system or presume that the overall uncertainties or its time derivative are norm bounded by a constant. Due to unmodelled dynamics and modelling imperfection, true structural knowledge of the system is not always available. Further, for the class of systems under consideration, prior assumption regarding the uncertainties (or its time derivative) being upper bounded by a constant, puts a restriction on states beforehand. Conventional ARC laws invite overestimation-underestimation problem of switching gain. Towards this front, Adaptive Switching-gain based Robust Control (ASRC) is proposed which alleviates the overestimation-underestimation problem of switching gain. Moreover, ASRC avoids any presumption of constant upper bound on the overall uncertainties and can negotiate uncertainties regardless of being linear or nonlinear in parameters. Experimental results of ASRC using a wheeled mobile robot notes improved control performance in comparison to adaptive sliding mode control.

I. INTRODUCTION

Adaptive and robust control address different costs of uncertainty handling, while existing ARC approaches retain parameter-bound or uncertainty-bound assumptions. For EL systems, constant uncertainty bounds can restrict states a priori, motivating a dedicated ARC framework.

A. Background

  • Adaptive control computes unknown parameters and gains online, while robust control reduces computation but requires a predefined uncertainty bound that may be unavailable under unmodelled dynamics.Assuming larger bounds to expand the operating region can increase switching gain and control effort.Source: A. Background · p. 1
  • ARC methods using projection operators require lower and upper bounds on individual parameters, while other approaches assume constant bounds on overall uncertainty or its derivative.Some adaptive laws also risk switching-gain overestimation or underestimation around a predefined threshold -.Source: A. Background · p. 1

B. Motivation

  • Although EL-system uncertainties may be nonlinear in parameters, their overall upper bound can have a linear-in-parameters structure.The chemostat example illustrates nonlinear-in-parameters uncertainties with linear-in-parameters upper bounds.Source: B. Motivation · p. 1
  • For EL systems, assuming a constant bound on overall uncertainty or its derivative restricts system states a priori, and prior parameter bounds may also be unavailable.EL systems span robotic manipulators, mobile robots, ships, aircraft, and other applications.Source: B. Motivation · p. 2

C. Contribution

The paper proposes ASRC for tracking uncertain Euler-Lagrange systems by using a linear-in-parameters structure in the uncertainty bound. It targets switching-gain estimation issues and is experimentally compared with ASMC on a wheeled mobile robot.

D. Contribution

  • ASRC is proposed for tracking uncertain EL systems whose uncertainty upper bound has a linear-in-parameters structure.It accommodates uncertainties that are either linear or nonlinear in parameters without assuming a constant upper bound on overall uncertainty or its time derivative.Source: C. Contribution · p. 2
  • The adaptive law allows switching gain to decrease when tracking error decreases, addressing switching-gain overestimation and underestimation.The formulation is described as insensitive to whether uncertainties are linear or nonlinear in parameters.Source: C. Contribution · p. 2
  • The paper evaluates ASRC against ASMC experimentally using a PIONEER 3 wheeled mobile robot.Source: C. Contribution · p. 2

D. Organization and Notations

The paper outlines the progression from its ASRC framework to experiments and concluding remarks. It also defines notation for minimum eigenvalues, norms, identity matrices, and positive real numbers.

D. Organization and Notations

  • Section II details the proposed ASRC framework, Section III reports experiments and comparison with ASMC, and Section IV gives concluding remarks.Source: D. Organization and Notations · p. 2
  • The notation defines λmin(·) as minimum eigenvalue, ||·|| as Euclidean norm, I as an identity matrix, and R+ as positive real numbers.Source: D. Organization and Notations · p. 2

A. Problem Formulation

The problem formulation models second-order EL dynamics with bounded disturbances and specified system properties. It defines a filtered tracking error and characterizes the resulting overall uncertainty through a linear-in-parameters upper bound.

A. Problem Formulation

  • The second-order EL model includes inertia, Coriolis and centripetal terms, gravity, slip/damping/friction forces, disturbances, and generalized control input.The mass matrix is uniformly positive definite, and the model specifies bounds on gravity, velocity-scaled friction, and disturbance.Source: A. Problem Formulation · p. 2
  • The desired trajectory and its first two derivatives are bounded, and the tracking error is defined relative to that trajectory.The filtered tracking error is introduced for the control formulation.Source: A. Problem Formulation · p. 2
  • The filtered-error dynamics isolate an overall uncertainty term that includes model terms, desired acceleration, and external disturbance.The paper defines this lumped term as σ in the filtered-error equation.Source: A. Problem Formulation · p. 2
  • Using the system properties and bounded desired trajectory, the upper bound on σ is expressed with linear-in-parameters coefficients.The passage identifies positive coefficients θ∗_i for i = 0, 1, 2 and defines estimates constrained to be at least those coefficients.Source: A. Problem Formulation · p. 2
  • Evaluating the robust switching gain conservatively requires knowledge of Θ∗, while constant bounds on uncertainty or its derivative restrict states a priori.The cited existing switching-gain approach is also described as suffering overestimation and underestimation.Source: A. Problem Formulation · p. 3

B. Adaptive Switching-gain based Robust Control (ASRC)

ASRC uses the LIP structure of the uncertainty upper bound to control uncertain EL systems even when the underlying uncertainties are LIP or NLIP. Its adaptive switching gains can decrease as well as increase, with a theorem bounding when decrease begins.

B. Adaptive Switching-gain based Robust Control (ASRC)

  • ASRC compensates LIP or NLIP uncertainties by exploiting the LIP structure of their upper bound.The proposed framework is for tracking control of uncertain Euler-Lagrange systems.Source: B. Adaptive Switching-gain based… · p. 3
  • The robust switching gain is parameterized as ρ̂ = θ̂0 + θ̂1||ξ|| + θ̂2||ξ||^2 + γ, with γ an auxiliary gain and adaptive laws updating the gains.The robust term Δτ uses ρ̂ to counter σ.Source: B. Adaptive Switching-gain based… · p. 3
  • For ||ef|| < ϖ, the tracking error remains inside Bϖ and the gains stay unchanged; reducing ϖ tightens this region provided chattering is avoided.Source: B. Adaptive Switching-gain based… · p. 3
  • After gains begin increasing, all gains begin decreasing within a finite interval T ≤ t̄ + δt.The theorem bounds the interval from the start of gain increase to the start of decrease.Source: B. Adaptive Switching-gain based… · pp. 3–4
  • Once the finite transient passes, the Lyapunov function decays exponentially, supporting the proof that gains eventually begin decreasing.The stated decay is V̇ ≤ −ϱV for t ≥ tin + t̄.Source: B. Adaptive Switching-gain based… · p. 4

C. Stability Analysis

The stability analysis establishes bounded closed-loop behavior across three adaptation cases and characterizes how the positive root of a polynomial bounds tracking behavior. ASRC’s gain rules address the overestimation and underestimation associated with the ASMC switching gain.

Stability Analysis

  • Theorem 2 guarantees uniform ultimate boundedness of the tracking errors, filtered errors, parameter-estimation errors, and auxiliary gain.The common Lyapunov analysis covers the three adaptation cases and establishes UUB for the overall closed-loop system.Source: C. Stability Analysis · pp. 4–5
  • The adaptive gains remain bounded when errors exceed the threshold and increase, decrease, or stay constant across the three analyzed cases.The cases are ||ef|| ≥ ϖ with gains increasing or decreasing, and ||ef|| < ϖ with parameter estimates and γ held constant.Source: C. Stability Analysis · pp. 4–5
  • The polynomial has exactly one positive real root ι, and the system is UUB when ||ξ|| ≥ ι.The root values need not be determined; increasing ς reduces ι and can improve tracking accuracy.Source: C. Stability Analysis · p. 5
  • Condition (20) is necessary for stability, and excessively high ς may trigger γ ≤ β and increase all adaptive gains.The analysis-only scalars are not used to design the control law.Source: C. Stability Analysis · p. 5
  • The auxiliary gain γ speeds adaptation when it increases and contributes a negative fourth-degree term that supports stability when it decreases.For systems without a Coriolis term, the analysis uses a corresponding linear-in-||ξ|| uncertainty-bound structure and a cubic adaptive-law term.Source: C. Stability Analysis · pp. 5–6
  • ASRC decreases gains as errors approach zero above ϖ and holds sufficient gains constant inside Bϖ, alleviating ASMC’s overestimation and underestimation problems.Reducing ϖ can improve tracking accuracy as long as chattering does not occur.Source: C. Stability Analysis · p. 6

III. APPLICATION: NONHOLONOMIC WMR

The section applies ASRC to a nonholonomic wheeled mobile robot (WMR), whose practical uncertainties include friction, slip, skid, and external disturbances. It motivates robust tracking and explains that ASRC does not require the robot’s dynamics or uncertainty bounds to be known.

III. APPLICATION: NONHOLONOMIC WMR

  • The application compares ASRC and adaptive sliding mode control on a commercial PIONEER 3 robot tracking a desired WMR trajectory.The reduced-order robot model includes unmodelled dynamics and disturbance terms.Source: §III · pp. 6–7
  • The WMR has five generalized coordinates but only two control inputs, so its dynamics are represented by reduced-order dynamics and a kinematic model.The wheels’ positions are directly controllable, unlike the robot’s center-of-mass coordinates and heading.Source: §III · p. 7
  • The model accounts for unmodelled friction and disturbances alongside nominal dynamics that assume rolling without slipping.Payload changes can also vary system mass, center of mass, and inertia; prior-bound switching-gain evaluation would require knowledge of these variations and uncertainty bounds.Source: §III · p. 7
  • ASRC adapts its uncertainty-bound parameters without requiring knowledge of the WMR dynamics, unmodelled terms, disturbances, or their time-varying bounds.The coordinate transformation matrix S(q) is used to represent WMR pose, not to design the control law.Source: §III · p. 7

A. Experimental Scenario

The experiment directs the PIONEER 3 WMR along a circular path while periodically adding and removing payload. Encoder-based measurements and a 20 ms sampling interval support implementation of the compared controllers.

A. Experimental Scenario

  • The WMR is directed to follow a circular path using specified desired trajectories.The robot’s desired and actual pose are determined using wheel velocities, with actual velocities obtained from encoders.Source: A. Experimental Scenario · p. 7
  • The PIONEER 3 uses 500 ppr incremental encoders, starts with both wheel positions at zero, and uses an initial wheel-position error of (π/10, π/10) rad.All controllers use a 20 ms sampling interval; their parameters are specified separately for ASRC and ASMC.Source: A. Experimental Scenario · pp. 7–8
  • A 3.5 kg payload is added and removed periodically, remaining on or off the robot for five seconds at a time.The payload is varied at different places on the robotic platform.Source: A. Experimental Scenario · pp. 7–8

B. Experimental Results and Comparison

On the wheeled mobile robot, ASRC tracks the desired circular path more accurately than ASMC. Its adaptive gains avoid the over- and underestimation patterns described for ASMC, including when ASRC starts with lower gains.

B. Experimental Results and Comparison

  • ASMC’s switching gain rises as ||s|| approaches zero, then can become insufficient; ASRC gains decrease with error and hold when sufficient.The paper associates ASMC’s behavior with overestimation and later underestimation, while ASRC avoids both patterns.Source: B. Experimental Results and Comparison · p. 8, Figure 5 · p. 8, Figure 6 · p. 9
  • With reduced ϖ, ASRC tracking accuracy improves while chattering is not observed in the control input.Table II reports performance for various ϖ; other control parameters are kept unchanged.Source: Table II · p. 8
  • ASRC provides better tracking accuracy than ASMC on the wheeled mobile robot.Figure 4 compares the controllers; the reported metrics are position error Ep and control effort Eτ.Source: Figure 4 · p. 8, B. Experimental Results and Comparison · p. 8
  • Starting ASRC with gains of 10 initially yields higher tracking error, but its performance becomes similar to ASRC1 from t ≥12 sec and remains much improved over ASMC.ASRC1 starts with gains of 20; ASRC2 starts with gains of 10.Source: B. Experimental Results and Comparison · p. 9, Figure 8 · p. 9
  • For ASRC2, gains remain unchanged once they suffice to keep filtered tracking error within ϖ, avoiding over- and underestimation despite low initial gains.The gains do not increase while ||eR|| decreases during t = 3.78−8 sec, and remain unchanged from t ≥8 sec.Source: B. Experimental Results and Comparison · p. 9, Figure 9 · p. 9
  • Under the first conditions of (46)–(47), θ̂0 and γ increase at the same rate and have similar values in Fig. 9 as ||eR|| increases during t = 0–3.7 sec; their falling rates differ under the second conditions.The experiment selected α0 = α3 = 10.Source: B. Experimental Results and Comparison · p. 9

IV. CONCLUSIONS

The paper proposes ASRC for uncertain Euler–Lagrange systems whose uncertainty upper bound has a LIP structure. It reports that the method handles LIP or NLIP uncertainties and alleviates switching-gain over- and underestimation.

IV. CONCLUSIONS

  • ASRC is proposed for uncertain Euler–Lagrange systems whose uncertainty upper bound has a LIP structure.The method is described as independent of whether the uncertainties themselves are LIP or NLIP.Source: §IV · p. 10
  • ASRC avoids assuming a constant upper bound on overall uncertainty and avoids a prior restriction on system states.These are stated advantages of the proposed adaptive-robust control law.Source: §IV · p. 10
  • Experiments validate ASRC against adaptive sliding mode control, and extending the law to systems with unmatched disturbances is future work.Source: §IV · p. 10
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