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Upper Bounding Estimation for Robustness to the Parameter Uncertainty with Trigonometric Function in Trajectory Control of Robot Arms

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Abstract

In this paper, a new robust control law is considered for controlling robot manipulators subjected to uncertainties. The control law is derived as a result of analytical solution from the Lyapunov function, thus stability of the uncertain system is guaranteed. Apart from previous studies, uncertainty bound and adaptation gain matrix are updated in time with the estimation law to control the system properly and uncertainty bound is determined using a trigonometric function of robot kinematics, inertia parameters and tracking error while adaptation gain matrix is determined using a trigonometric function of robot kinematics and tracking error. Application to a two-link robotic manipulator is presented and numerical simulations are included.

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Abbreviations

q, \(\dot{q}\) and \(\ddot{q}\), ɛR n :

Vectors of joint position, velocity and acceleration of robot

M(q) ɛR n×n, \(C(q,\ \dot{q})\dot{q}\) and G(q), ɛR n :

Inertia matrix, centripetal/coriolis and gravitational vectors

q d, \(\dot{q}_{d}\) and \(\ddot{q}_{d}\), ɛR n :

Desired position, velocity and acceleration vectors

\(\tilde{q} = q_{d} - q,\;\dot{{\tilde{q}}} = \dot{q}_{d} - \dot{q},\;\varepsilon R^{n}\) :

Actual position and velocity errors

\(Y(q,\ \dot{q},\ \ddot{q}),\;\varepsilon R^{{nxp}}\) :

A matrix which is a function of joint positions, velocities and accelerations

\(Y(q,\ \dot{q},\ \dot{q}_{r},\ \ddot{q}_{r})\varepsilon R^{{nxp}}\) :

A control action of inverse dynamics type which ensures an approximate compensation of nonlinear effects and joint decoupling

Λ, K, and B ɛR n×n :

Positive definite diagonal matrixes

K σ ɛR n :

A vector of PD action

ɛ > 0:

A positive number

π 0 ɛR p :

A fixed vector of nominal, loaded arm parameters and their upper bounds

ρ ɛR :

Parametric uncertainty

\(\hat{\rho }(t)\varepsilon R^{p}\) :

Estimated upper bound

α, β, γ :

Real numbers

γ 1, γ 2, ... γ p :

Real numbers

where λ 1, λ 2, .... λ p and α 1, α 2, ....α p :

Real numbers

a, b ɛR :

Real numbers

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  1. Vocational College of Kayseri, Erciyes University, 38039, Kayseri, Turkey

    Recep Burkan

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Burkan, R. Upper Bounding Estimation for Robustness to the Parameter Uncertainty with Trigonometric Function in Trajectory Control of Robot Arms. J Intell Robot Syst 46, 263–283 (2006). https://doi.org/10.1007/s10846-006-9061-5

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  • DOI: https://doi.org/10.1007/s10846-006-9061-5

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