A family of virtual contraction based controllers for tracking of flexible‐joints port‐Hamiltonian robots: Theory and experiments

Abstract

In this work, we present a constructive method to design a family of virtual contraction based controllers that solve the standard trajectory tracking problem of flexible-joint robots in the port-Hamiltonian framework. The proposed design method, called virtual contraction based control, combines the concepts of virtual control systems and contraction analysis. It is shown that under potential energy matching conditions, the closed-loop virtual system is contractive and exponential convergence to a predefined trajectory is guaranteed. Moreover, the closed-loop virtual system exhibits properties such as structure preservation, differential passivity, and the existence of (incrementally) passive maps. The method is later applied to a planar RR robot, and two nonlinear tracking control schemes in the developed controllers family are designed using different contraction analysis approaches. Experiments confirm the theoretical results for each controller. K E Y W O R D Scontraction, flexible-joints robots, port-Hamiltonian systems, tracking control, virtual control systems INTRODUCTIONControl problems in rigid robots have been widely studied in the literature due to they are instrumental in modern manufacturing systems. However, as pointed out in Nicosia and Tomei 1 the elasticity in the joints often cannot be neglected for accurate position tracking. For every joint, that is, actuated by a motor, we basically need two degrees of freedom (dof) instead of one. Such flexible-joint robots (FJRs) are therefore underactuated mechanical systems. In the work of Spong 2 two state feedback control laws based, respectively, on feedback linearization and singular perturbation theory are presented for a simplified FJRs model. Similarly, in de Wit et al 3 a dynamic feedback controller for a more detailed model is presented. In Loría and Ortega 4 a computed-torque controller for FJRs is designed, which does not need jerk measurements. In Ailon and Ortega 5 and Brogliato et al 6 passivity-based control (PBC) schemes are proposed. The first one is an observer-based controller, which requires only motor position measurements. In the latter one, a PBC controller is designed and compared with backstepping and decoupling techniques. For further details on PBC of FJRs, we refer Partial results were presented in the IFAC Workshop on Lagrangian and Hamiltonian Methods in Nonlinear Control 2018. Int J Robust Nonlinear Control. 2020;30:3269-3295. wileyonlinelibrary.com/journal/rnc

Nonlinear parameter‐varying state‐feedback design for a gyroscope using virtual control contraction metrics

“…Integration of the above dissipation condition along the geodesics gives the universal  2 -gain bound of 𝛼. 16 Similar to the case of H ∞ state-feedback control for linear systems, 38 the condition (24) can be converted into the following pointwise LMI:…”

“…Some recent works include control synthesis for a special case of mechanical systems 22 and further extension to port-Hamiltonian systems. 23,24 The main idea of virtual systems is that a NL system, which is not itself contracting, may have weaker stability properties that can be established via construction of an auxiliary (virtual) system which is contracting. Furthermore, the virtual system can be seen as an NPV embedding of the dynamics of the original system.…”

Contraction Analysis of Discrete-time Stochastic Systems

“…for a given deterministic initial condition (k 0 , x k0 , ξ(k0−1)− ) ∈ Z × R n × Ξ(k0−1)− ; k 0 , x k0 , and ξ(k0−1)− denote the initial time, the initial state of the system (7), and the initial state of the stochastic process ξ, respectively. To emphasize that (x k ) k∈Z k 0 + : Ω → (R n ) Z k 0 + is a stochastic process under the initial condition (k 0 , x k0 , ξ(k0−1)− ) ∈ Z×R n × Ξ(k0−1)− , this is also denoted by (φ k (ξ (k−1)− ; k 0 , x k0 , ξ(k0−1)− )) k∈Z k 0 + or simply (φ k (ξ (k−1)− )) k∈Z k 0 + .…”

“…for the same deterministic initial time k 0 ∈ Z as the system (7) and a given deterministic initial state δx k0 ∈ R n of the variational system. From its definition, the variational system is also a stochastic system.…”

Neural‐based adaptive event‐triggered tracking control for flexible‐joint robots with random noises

“…Since flexible-joint robots can be widely used in industrial production to avoid the occurrence of dangerous situations, related research on the control of flexible-joint robots has very important theoretical and practical significance. [1][2][3][4] Moreover, there are varying degrees of uncertainty in the actual system, such as parameter uncertainty, unmodeled dynamics, and unknown disturbance. For nonlinear systems with these objective uncertainties, the adaptive control design plays an important role in constructing suitable controllers to achieve control objectives.…”