Optimal input design for flat systems using B-splines

Abstract

Abstract-This paper deals with optimal design of input signals for linear, controllable systems, by means of their flat output. The flat output is parametrized by a polynomial spline and a linear problem is formulated in which both the spline coefficients and the knot locations are found simultaneously. Conservative constraints on the spline coefficients ensure that semi-infinite bounds are never violated and numerical results show that the amount of conservatism is little.

Optimal Tube Following for Robotic Manipulators

“…As mentioned before, the bounds q i (s) and q i (s) should be chosen sufficiently large, such that the end effector tube constraint is the restricting constraints and not the joint bounds. To allow for varying p(s) along the path, we adopt the polynomial spline parametrisation, with B-spline basis, of degree l with g internal knots presented in (Van Loock et al, 2011).…”

Optimal trajectory planning and tracking control method for overhead cranes

“…Based on this characteristic, the trajectory planning problem of this kind of system can be successfully solved by planning the flat output, which has been widely used for the control of many mechanical systems [27–29]. On the other hand, B‐spline method has the advantages of good continuity and smoothness, which has been proven to be an effective tool for trajectory planning of various systems [30–32]. Inspired by these results, this paper proposes a novel trajectory planning and tracking scheme to construct an time‐optimal trolley trajectory with an analytical expression, as well as a non‐linear tracking controller, to achieve accurate cart positioning and high‐efficiency payload swing suppression.…”

“…Some simulation and experiments are performed in the environment of MATLAB/Simulink and on a self‐built crane test‐bed, respectively, with the results clearly showing the satisfactory performance of the proposed novel trajectory planning and tracking scheme. Compared with [32] and other related references, the contributions of the paper can be summarised as follows: (i) it constructs a time‐optimal trolley trajectory with sufficiently high transportation efficiency, which has an analytical expression and presents good continuity and smoothness characteristics, thus is convenient for tracking controller design; (ii) actual physical constraints, such as maximum permitted swing angle, actuator limits and so on, are fully considered when constructing the desired trajectory; and (iii) as supported by experimental results, the planned trajectory is successfully tracked by the designed non‐linear controller.…”

Minimum-Time Trajectory Planning for Underactuated Overhead Crane Systems With State and Control Constraints

“…Indeed, these aforementioned works successfully yield several (robust or nonrobust) minimum-time control techniques satisfying the boundary conditions; however, they do not take into account the safety or physical constraints such as the limits of the swing angle and the trolley velocity during the motion process. Some minimum-time control methods also incorporate state constraints in the control design [21], [24]. The work in [24] converts the problem to a parametric optimization problem to obtain an optimal solution; however, the controller structure and the switching time are assumed to be empirically known, and hence, the consequent optimality is local, i.e., it only holds in a specified parametric space.…”

“…The work in [24] converts the problem to a parametric optimization problem to obtain an optimal solution; however, the controller structure and the switching time are assumed to be empirically known, and hence, the consequent optimality is local, i.e., it only holds in a specified parametric space. Loock et al utilized the property of differential flatness and fourth-order B-splines to optimize the trajectory in the parametric space [21]. Unfortunately, the velocity bounds are not taken into consideration, and moreover, the optimality is also local, which only holds in a fourth-order B-spline parametric space.…”