Generalized Cornu-type spirals and their Darboux parametric deformations

Rosu, H. C.; Mancas, Stefan C.; Hsieh, Chun Chung

doi:10.1016/j.physleta.2019.05.040

Cited by 5 publications

(7 citation statements)

References 7 publications

(15 reference statements)

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“…First, we must verify that the waypoints are set correctly (14) and exclude forbidden maneuvers for aircraft-type UAVs. Let us list the items that require verification:…”

Section: Primitive Trajectory Constructionmentioning

confidence: 99%

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Dynamic Models with Sigmoid Corrections to Generation of an Achievable 4D-Trajectory for a UAV and Estimating Wind Disturbances

2023

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For an unmanned aerial vehicle (UAV) of an aircraft type, the problems of planning achievable trajectories as well as robust control under wind disturbances are considered. A computationally simple method for compiling a primary non-smooth 4D trajectory is proposed. Its segments connect the given waypoints and determine the desired average velocity in various sections. Instead of time-consuming methods of analytical smoothing of broken path joints using polynomials, a tracking differentiator with S-shaped smooth and limited sigmoid corrective actions is developed. This virtual dynamic model provides natural smoothing of the primary trajectory considering the design constraints on the velocity, acceleration, and thrust of the UAV. The tracking differentiator variables create an achievable trajectory and are used to synthesize the UAV tracking system. To compensate for the action of wind disturbances on the UAV, a disturbance observer is developed. It is a replica of the equations of the control plant model, which are directly affected by external uncontrolled disturbances. These algorithms also use sigmoid corrections. Unlike standard disturbances observers, this approach does not require the development of a dynamic model of external disturbances and does not assume their smoothness. The effectiveness of the developed algorithms was confirmed by numerical simulation.

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“…First, we must verify that the waypoints are set correctly (14) and exclude forbidden maneuvers for aircraft-type UAVs. Let us list the items that require verification:…”

Section: Primitive Trajectory Constructionmentioning

confidence: 99%

“…They also often use B'ezier Curves and Bernstein polynomials, which provide continuous curvature in the presence of seven control waypoints. Efficient methods for obtaining smooth curves are polynomial splines [12], clothoids [13], generalized Cornu spirals [14], etc. Cubic B-splines [2,10,15] are the most widely used at present.…”

Section: Introductionmentioning

confidence: 99%

Dynamic Models with Sigmoid Corrections to Generation of an Achievable 4D-Trajectory for a UAV and Estimating Wind Disturbances

2023

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“…Sufficient conditions under which inequalities (25) will be correct at t ∈ [0; T], similarly to (14), have the form…”

Section: Designing a Three-block Tracking Differentiatormentioning

confidence: 99%

“…They allow one to obtain an analytical description of the trajectory and, consequently, its derivatives of any required order. To obtain smooth curves, they use fragments of circles, polynomial splines [12], clothoids [13], generalized Cornu spirals [14], etc. A typical modern solution is B-splines [15].…”

Section: Introductionmentioning

confidence: 99%

Generation of Achievable Three-Dimensional Trajectories for Autonomous Wheeled Vehicles via Tracking Differentiators

Krasnova,

Kokunko,

Kochetkov

et al. 2023

Algorithms

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Planning an achievable trajectory for a mobile robot usually consists of two steps: (i) finding a path in the form of a sequence of discrete waypoints and (ii) transforming this sequence into a continuous and smooth curve. To solve the second problem, this paper proposes algorithms for automatic dynamic smoothing of the primary path using a tracking differentiator with sigmoid corrective actions. Algorithms for setting the gains of the differentiator are developed, considering a set of design constraints on velocity, acceleration, and jerk for various mobile robots. When tracking a non-smooth primary path, the output variables of the differentiator generate smooth trajectories implemented by a mechanical plant. It is shown that the tracking differentiator with a different number of blocks also generates derivatives of the smoothed trajectory of any required order, taking into account the given constraints. Unlike standard analytical methods of polynomial smoothing, the proposed algorithm has a low computational load. It is easily implemented in real time on the on-board computer. In addition, simple methods for modeling a safety corridor are proposed, taking into account the dimensions of the vehicle when planning a polygon with stationary obstacles. Confirming results of numerical simulation of the developed algorithms are presented.

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“…The problem of admissible trajectory planning is a separate, rather time-consuming task. As a rule, it is solved off-line, using spline interpolation or complex geometric calculations [ 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 ]. If there is reason to believe that the given curves are smooth and realizable, then one can solve the problem of restoring their derivatives and filtering by the above methods (using dynamic differentiators with special tuning).…”

Section: Introductionmentioning

confidence: 99%

Dynamic Smoothing, Filtering and Differentiation of Signals Defining the Path of the UAV

Antipov

Kokunko

Krasnova

et al. 2022

Sensors

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On the example of a control system for an unmanned aerial vehicle, we consider the problems of filtering, smoothing and restoring derivatives of reference action signals. These signals determine the desired spatial path of the plant at the first approximation. As a rule, researchers have considered these problems separately and have used different methods to solve each of them. The paper aims to develop a unified approach that provides a comprehensive solution to mentioned problems. We propose a dynamic admissible path generator. It is constructed as a copy of the canonical control plant model with smooth and bounded sigmoid corrective actions. For the deterministic case, a synthesis procedure has been developed, which ensures that the output variables of the generator track a non-smooth reference signal. Moreover, it considers the constraints on the velocity and acceleration of the plant. As a result, the generator variables produce a naturally smoothed spatial curve and its derivatives, which are realizable reference actions for the plant. The construction of the generator does not require exact knowledge of the plant parameters. Its dynamic order is less than that of the standard differentiators. We confirm the effectiveness of the approach by the results of numerical simulation.

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Generalized Cornu-type spirals and their Darboux parametric deformations

Abstract: We generalize the Fresnel integrals and introduce a class of planar spirals Fn, which contains the Cornu spiral as the case F2. Their Darboux parametric deformations are also investigated. The F3 spiral and some of its Darboux deformed counterparts are graphically illustrated.

Cited by 5 publications

References 7 publications

Dynamic Models with Sigmoid Corrections to Generation of an Achievable 4D-Trajectory for a UAV and Estimating Wind Disturbances

Dynamic Models with Sigmoid Corrections to Generation of an Achievable 4D-Trajectory for a UAV and Estimating Wind Disturbances

Generation of Achievable Three-Dimensional Trajectories for Autonomous Wheeled Vehicles via Tracking Differentiators

Dynamic Smoothing, Filtering and Differentiation of Signals Defining the Path of the UAV

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