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

Three-dimensional Chained Nonholonomic Systems Stabilization Control via Dynamic Feedback

“…According to the definition of z, and equation (7), (9), (15), (16), there is the following equation: …”

Universal Practical Tracking Control of a Planar Underactuated Vehicle

“…We construct the input transformation to make become the second‐order nonholonomic chained form (see e.g . ) where the perturbed term D is …”

“…Note that s 3 ‐dynamic in is not in the desired form with a perturbed term, thus we introduce a new variable z as Differentiating the both sides of results in where …”

“…Unified/universal control schemes have been proposed in recent years to simultaneously achieve both the fixed-point stabilization and the trajectory tracking for second-order nonholonomic systems [5,6,[22][23][24] and other nonholonomic ones [25][26][27][28]. A negative result was provided in [29] revealing that the feedback laws, able to asymptotically stabilize any feasible reference trajectory, do not exist for nonholonomic systems, which is accordance with the results in [6,22,25,26] where the derived asymptotic tracking laws can only track the specific restricted feasible trajectories (a trajectory is called feasible if it can be generated by the control system dynamics; otherwise, it is non-feasible [28]). Moreover, any non-feasible trajectory can not be asymptotically stabilized either [27].…”

Simultaneous stabilization and trajectory tracking of underactuated mechanical systems with included actuators dynamics

“…second-order nonholonomic chained systems is given in [16], control of UNMS with servoconstraints is presented in [7][8][9]22], while control of holonomic and nonholonomic systems using sliding mode is presented by the direct Lyapunov approach [15]. Specific examples of simultaneous stabilization and/or trajectory tracking of UNMS are given in [3,14,27,33].…”