Semi-analytical minimum time solutions with velocity constraints for trajectory following of vehicles

Frego, Marco; Bertolazzi, Enrico; Biral, Francesco; Fontanelli, Daniele; Palopoli, Luigi

doi:10.1016/j.automatica.2017.08.020

Cited by 40 publications

(18 citation statements)

References 38 publications

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“…This approach has a three-level structure which combines triangular decomposition, Dijkstra's algorithm, and a proposed particle swarm optimization called constrained multi-objective particle swarm optimization. In Reference [28], to find an optimal maneuver that moves a car-like vehicle between two configurations in minimum time, a two-phase algorithm firstly solves a geometric optimization problem and then finds the optimal maneuver with system dynamics and its constraints satisfied.…”

Section: Hierarchical Path Planningmentioning

confidence: 99%

Combining Subgoal Graphs with Reinforcement Learning to Build a Rational Pathfinder

Zeng

Qin

et al. 2019

Applied Sciences

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In this paper, we present a hierarchical path planning framework called SG-RL (subgoal graphs-reinforcement learning), to plan rational paths for agents maneuvering in continuous and uncertain environments. By "rational", we mean (1) efficient path planning to eliminate first-move lags;(2) collision-free and smooth for agents with kinematic constraints satisfied. SG-RL works in a two-level manner. At the first level, SG-RL uses a geometric path-planning method, i.e., Simple Subgoal Graphs (SSG), to efficiently find optimal abstract paths, also called subgoal sequences. At the second level, SG-RL uses an RL method, i.e., Least-Squares Policy Iteration (LSPI), to learn near-optimal motion-planning policies which can generate kinematically feasible and collision-free trajectories between adjacent subgoals. The first advantage of the proposed method is that SSG can solve the limitations of sparse reward and local minima trap for RL agents; thus, LSPI can be used to generate paths in complex environments. The second advantage is that, when the environment changes slightly (i.e., unexpected obstacles appearing), SG-RL does not need to reconstruct subgoal graphs and replan subgoal sequences using SSG, since LSPI can deal with uncertainties by exploiting its generalization ability to handle changes in environments. Simulation experiments in representative scenarios demonstrate that, compared with existing methods, SG-RL can work well on large-scale maps with relatively low action-switching frequencies and shorter path lengths, and SG-RL can deal with small changes in environments. We further demonstrate that the design of reward functions and the types of training environments are important factors for learning feasible policies.

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Section: Hierarchical Path Planningmentioning

confidence: 99%

Combining Subgoal Graphs with Reinforcement Learning to Build a Rational Pathfinder

Zeng

Qin

et al. 2019

Applied Sciences

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“…We specify exactly the BCs of problem P3, that is, we fix the initial point, angle and curvature, or ( x 0 , y 0 ), θ 0 and κ 0 , and the final point ( x 1 , y 1 ); the algorithm to solve P3 finds

κ^{′}

and the length L , such that the connection exhibits G 2 continuity. There is then a slave algorithm that computes the optimal speed profile of the obtained spline and returns the value to the master algorithm that decides whether this trajectory is good or not; eventually, the segment is rejected and another point is tested. We did a numeric simulation of the time‐optimal path on the tracks of Monza and Silverstone (see Figure ) with the optimal control solver PINS .…”

Section: Numerical Testsmentioning

confidence: 99%

“…In this case, the solver is slowed down or does not converge at all. Restriction (11) is not a real limitation, however. The parameters passed to IPOPT are a tolerance of 10 −10 , the analytic gradient of the constraints, and the Hessian approximated with the limited memory BFGS.…”

Section: Numerical Testsmentioning

confidence: 99%

Interpolating clothoid splines with curvature continuity

Bertolazzi

Frego

2017

Math Methods in App Sciences

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An efficient algorithm for the computation of a C2 interpolating clothoid spline is herein presented. The spline is obtained following an optimisation process, subject to continuity constraints. Among the 9 various targets/problems considered, there are boundary conditions, minimum length path, minimum jerk, and minimum curvature (energy). Some of these problems are solved with just a couple of Newton iterations, whereas the more complex minimisations are solved with few iterations of a nonlinear solver. The solvers are warmly started with a suitable initial guess, which is extensively discussed, making the algorithm fast. Applications of the algorithm are shown relating to fonts, path planning for human walkers, and as a tool for the time‐optimal lap on a Formula 1 circuit track.

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“…All these blocks are used inside a high‐level optimiser that constructs the trajectory of the carlike vehicle along a sequence of given points. ()…”

Section: Introductionmentioning

confidence: 99%

“…All these blocks are used inside a high-level optimiser that constructs the trajectory of the carlike vehicle along a sequence of given points. 4,27,28 In this work, we focus on the fast and accurate numerical computation of the analytic solution of an OCP arising from a Riccati dynamical system. We remark, since the limit of the lateral acceleration is a function of the state only (and not of the control variable) that we can solve the complete problem in 2 phases.…”

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confidence: 99%

Semianalytical minimum‐time solution for the optimal control of a vehicle subject to limited acceleration

Bertolazzi

Frego

2017

Optim Control Appl Methods

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Summary A semianalytic solution of the minimum‐time optimal control problem of a carlike vehicle is herein presented. The vehicle is subject to the effect of laminar (linear) and aerodynamic (quadratic) drag, taking into account the asymmetric bounded longitudinal accelerations. This yields a nonlinear differential equation of the Riccati kind. The associated analytic solution exists in closed form, but it is numerically ill conditioned in some situations; hence, it is reformulated using asymptotic expansions that keep the numerical computations accurate and robust in all cases. Therefore, the proposed algorithm is designed to be fast and robust in sight to be the fundamental module of a more extended optimal control problem that considers a given clothoid curve as the trajectory and the presence of a constraint on the lateral acceleration of the vehicle. The numerical stability of the computation for limit values of the parameters is essential, as shown in the numerical tests.

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Semi-analytical minimum time solutions with velocity constraints for trajectory following of vehicles

Cited by 40 publications

References 38 publications

Combining Subgoal Graphs with Reinforcement Learning to Build a Rational Pathfinder

Combining Subgoal Graphs with Reinforcement Learning to Build a Rational Pathfinder

Interpolating clothoid splines with curvature continuity

Semianalytical minimum‐time solution for the optimal control of a vehicle subject to limited acceleration

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