Cooperative Trajectory Generation Using Pythagorean Hodograph Bézier Curves

Choe, Ronald; Puig-Navarro, Javier; Cichella, Venanzio; Xargay, Enric; Hovakimyan, Naira

doi:10.2514/1.g001531

Cited by 53 publications

(35 citation statements)

References 34 publications

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“…where the last equality follows from Equation (28). An identical argument can be used to show that the closure condition (35) holds, thus completing the proof of Theorem 3. ♠ Corollary 2 If solutions x * (t), u * (t), λ * (t), µ * (t) and ν * of Problem P λ exist and satisfyẋ * (t) ∈ C 2 nx , u * (t) ∈ C 2 nu ,λ * (t) ∈ C 2 nx , and µ * (t) ∈ C 2 n h in [0, 1], then Theorem 3 holds with δ N P = C P N −1 and δ N D = C D N −1 , where C P and C D are positive constants independent of N .…”

Section: Kkt Conditions Of Problem P Nmentioning

confidence: 74%

“…In what follows, we show that the above polynomials satisfy the constraints in (11), (12) and (13), with δ N P defined in Equation (15), thus proving Theorem 1. Equations (16) and (17), together with Lemma 1 and Assumption 2, imply that x N (t),ẋ N (t) and u N (t) converge uniformly to x(t),ẋ(t) and u(t), respectively. More precisely, for all t ∈ [0, 1] from Lemma 1 we have…”

Section: Feasibility and Consistency Of Problem P Nmentioning

confidence: 99%

“…Furthermore, we notice that if x ∞ (t), u ∞ (t),λ ∞ (t),μ ∞ (t) andν ∞ satisfy Equations (33)- (35), then the following holds for all k = 0, . .…”

Section: Kkt Conditions Of Problem P Nmentioning

confidence: 99%

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Consistent approximation of optimal control problems using Bernstein polynomials

Cichella

Kaminer

Walton

et al. 2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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Bernstein polynomial approximation to a continuous function has a slower rate of convergence as compared to other approximation methods. "The fact seems to have precluded any numerical application of Bernstein polynomials from having been made. Perhaps they will find application when the properties of the approximant in the large are of more importance than the closeness of the approximation." -has remarked P.J. Davis in his 1963 book Interpolation and Approximation.This paper presents a direct approximation method for nonlinear optimal control problems with mixed input and state constraints based on Bernstein polynomial approximation. We provide a rigorous analysis showing that the proposed method yields consistent approximations of time continuous optimal control problems. Furthermore, we demonstrate that the proposed method can also be used for costate estimation of the optimal control problems. This latter result leads to the formulation of the Covector Mapping Theorem for Bernstein polynomial approximation. Finally, we explore the numerical and geometric properties of Bernstein polynomials, and illustrate the advantages of the proposed approximation method through several numerical examples.

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Section: Kkt Conditions Of Problem P Nmentioning

confidence: 74%

Section: Feasibility and Consistency Of Problem P Nmentioning

confidence: 99%

See 1 more Smart Citation

Consistent approximation of optimal control problems using Bernstein polynomials

Cichella

Kaminer

Walton

et al. 2019

2019 IEEE 58th Conference on Decision and Control (CDC)

Self Cite

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“…• Trajectory generation algorithm: 23,24 is responsible for the computation of a geometrical path and a desired speed profile v d,i (t) for each of the n vehicles in the fleet, i ∈ I := {1, . .…”

Section: Background: Time-critical Cooperative Path Followingmentioning

confidence: 99%

“…Since the reference agent runs independently of all other vehicles, the time evolution along its path is exclusively determined by the reference rateẋ R (t) and the desired speed profile v d,R (t), computed during the trajectory generation phase. 23 As a result, each point along the path of the reference agent has a fixed temporal specification, as highlighted in Figure 2. It is remarked thatẋ R (t) = 1 orẋ i (t) = 1 for all t does not imply that the speed profile of the corresponding vehicle is constant.…”

Section: Assumptionmentioning

confidence: 99%

Time-Coordination Strategies and Control Laws for Multi-Agent Unmanned Systems

Puig-Navarro

Hovakimyan²,

Allen

2017

17th AIAA Aviation Technology, Integration, and Operations Conference

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Time-critical coordination tools for unmanned systems can be employed to enforce the type of temporal constraints required in terminal control areas, ensure minimum distance requirements among vehicles are satisfied, and successfully perform coordinated missions. In comparison with previous literature, this paper presents an ampler spectrum of coordination and temporal specifications for unmanned systems, and proposes a general control law that can enforce this range of constraints. The constraint classification presented considers the nature of the desired arrival window and the permissible coordination errors to define six different types of time-coordination strategies. The resulting decentralized coordination control law allows the vehicles to negotiate their speeds along their paths in response to information exchanged over the communication network. This control law organizes the different members in the fleet hierarchically per their behavior and informational needs as reference agent, leaders, and followers. Examples and simulation results for all the coordination strategies presented demonstrate the applicability and efficacy of the coordination control law for multiple unmanned systems.Recent developments in unmanned systems (UxSs) technologies have opened vast opportunities in scientific research, 1-3 remote monitoring and control, 4 rapid assessment of catastrophic events, 5 search and rescue operations, 6, 7 cartography, 8, 9 and cinematography, to name a few examples. Not only have UxSs eliminated the constraints and limitations of manned vehicles, but also their relatively low cost, miniaturization levels, and ease of deployment have popularized their use in the aforementioned areas. Moreover, fleets of cooperating UxSs can be used to exploit synergistic behaviors among vehicles, remove the risk of single-point failures, improve flexibility, increase redundancy and reliability, achieve blanket coverage of larger areas, and often reduce mission execution times.A fundamental component to achieve these goals is the availability of cooperative control strategies that are robust to temporary communication dropouts, packet loss, and external disturbances. Additionally, these cooperative strategies shall ensure safe inter-vehicle separation, while being able to adapt to changing mission goals, fleet size, and vehicle diversity. Technical advances in time coordination have the potential to realize new transportation paradigms that hold promise for revolutionary changes in other fields such as air traffic management, 10 mail services, and ground transportation systems.Inspired by these challenges, previous work 11, 12 developed a set of algorithms for time-critical cooperative path-following control of a fleet of Unmanned Aerial Vehicles (UAVs) that exchange information over a communication network. The resulting framework divides the fleet into leaders and followers and guarantees that the UAVs follow their assigned trajectories, while meeting temporal and spatial constraints. Moreover,

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New Developments in Theory, Algorithms, and Applications for Pythagorean–Hodograph Curves

Farouki

Giannelli

Sestini

2019

Advanced Methods for Geometric Modeling and Numerical Simulation

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This series will publish textbooks, multi-authors books, thesis and monographs in English language resulting from workshops, conferences, courses, schools, seminars, doctoral thesis, and research activities carried out at INDAM -Istituto Nazionale di Alta Matematica, http://www.altamatematica.it/en. The books in the series will discuss recent results and analyze new trends in mathematics and its applications.

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Cooperative Trajectory Generation Using Pythagorean Hodograph Bézier Curves

Cited by 53 publications

References 34 publications

Consistent approximation of optimal control problems using Bernstein polynomials

Consistent approximation of optimal control problems using Bernstein polynomials

Time-Coordination Strategies and Control Laws for Multi-Agent Unmanned Systems

New Developments in Theory, Algorithms, and Applications for Pythagorean–Hodograph Curves

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