Stabilization of underactuated mechanical systems: A non-regular back-stepping approach

Abstract

This paper presents a design framework for the stabilization of a class of underactuated mechanical systems. By utilizing nonregular static state feedbacks, these systems are transformed into a class of nonlinear systems with chain structures. Then, controller design is presented by applying the backstepping design technique. The design procedure is applied to an underactuated robotic system and simulation tests are carried out for illustrating the effectiveness of the proposed approach.

Unified controller for both trajectory tracking and point regulation of second-order nonholonomic chained systems

“…The second-order nonholonomic chained form (1) is not pure state feedback stabilizable according to Brokett's necessary condition in [17], however, the control law (11) and (12) is in the form of dynamic state feedback, and does not contradict the Brokett's necessary condition.…”

“…Different from the general back-steeping scheme where the control laws for true inputs was designed to track only virtual input signals, control law (12) is special in that it is designed to track both the virtual input signal v 2d and its integral y 2d = y 2d (0) + t 0 v 2d (τ )dτ . This special property is crucial to assure the convergences of tracking errors.…”

“…Xu et al [11] converted a class of second-order nonholonomic systems to the second-order nonholonomic chained form, and proposed a discontinuous control law to stabilize states to the desired equilibrium point exponentially. Sun et al [12] presented a design framework for the stabilization of a class of underactuated mechanical systems based on nonregular static state feedbacks approach. In [7], a class of underactuated systems was first converted to secondorder nonholonomic chained form by constructing the input and coordinate transformations explicitly, and then a smooth time-varying control scheme is proposed to stabilize all the states to zero exponentially.…”

Universal Practical Tracking Control of a Planar Underactuated Vehicle

“…] , > 1. Note that the ultimate bounds of (e 1 , e 2 , e 3 , e 1 ,ė 3 ) can be made arbitrarily small, while that oḟ e 2 can not, mainly due to the un-adjustable terms (a 1 , a 2 , b 1 ) in the expression of 12 . If (a 1 , a 2 , b 1 ) equal zero, i.e.,̇d =̈d = 0,ẍ 2 d +ÿ 2 d = 0, then 12 = k 2 + 4k 1 + 8 2 k −1 3 3.5 + 14k 3 can be reduced at will, implying that all the tracking errors ( e 1 , e 2 , e 3 , e 1 ,ė 2 ,ė 3 ) are UGPAC and the whole error system (27) is UGPAS on K.…”

“…For nonholonomic systems, due to the nonexistence of static smooth time-invariant state feedback control laws achieving the asymptotic point stabilization [8], the control objectives are usually classified to two categories: fixed-point stabilization [9][10][11][12][13][14][15][16][17] and trajectory tracking Manuscript [4,[18][19][20][21]. The controllers developed for the two control objectives are effective in many practical situations, but usually not suitable for some complicated control goals, for example the docking of vehicles that consists of the both tasks.…”