A predictive neural hierarchical framework for on-line time-optimal motion planning and control of black-box vehicle models

Abstract

This paper addresses the on-line minimum-time motion planning and control of a black-box racing vehicle model. We present a hierarchical control framework, composed of a high-level non-linear model predictive controller (NMPC) based on an advanced kineto-dynamical vehicle model, a low-level neural network to compute the inverse steering dynamics and a longitudinal controller for the low-level tracking of speed profiles. An off-line identification procedure, consisting of simulated manoeuvres, is defined to learn the high-level and low-level models. A closed-loop simulation is setup to control the black-box vehicle near the limits of handling along a racetrack. Simulation results are compared with the off-line solution of a minimum-time-optimal control problem.

Keywords:

• Autonomous racing
• model-predictive control
• motion planning
• identification of vehicle dynamics
• neural networks

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Other techniques exist to handle the inequality constraints, such as slack variables [59,60]. However, for the NMPC formulation of this paper (Section 4.1), penalty functions result in a good numerical efficiency, and we do not employ the slack reformulation.

2 AnteMotion Srl website: https://antemotion.com.

3 Such local violations may occur due to some mismatch between the predicted dynamics and the real behaviour of the vehicle model.

4 ${\zeta }_i$ is the curvilinear abscissa of the vehicle centre of mass (measured along the circuit middle line) at the moment of calculating a new MPC solution.

5 Polynomials of 2nd–3rd order are usually enough, at least for an electric powertrain.

6 The high-level NMPC model with the speed-dependant handling diagram is used only for a comparison between an extension of [6] and the NMPC formulation proposed in this paper (Section 4.1).

7 The windows of past and future (predicted) state profiles span $T_{{\mathrm{s}}_{\mathrm{N}\mathrm{N}}}\cdot n_{\mathrm{p}}=1.5$ s and $T_{{\mathrm{s}}_{\mathrm{N}\mathrm{N}}}\cdot n_{\mathrm{f}}=1$ s, respectively, which is enough for the time scale of the lateral and longitudinal vehicle dynamics.

8 We propose to select $N_{\mathrm{v}}$ so that the spacing among the speed values in the list is around 10–15 km/h.

9 Equation (15(15) $\delta \left(t\right)-\frac{{\mathrm{\Omega }}_{ss}\left(t\right)}{v_x\left(t\right)}L=K_{us}\left(a_y\right(t),v_x(t\left)\right)⟶{\mathrm{\Omega }}_{ss}\left(t\right)=v_x\left(t\right)\left[\frac{\delta \left(t\right)-K_{us}\left(a_y\right(t),v_x(t\left)\right)}{L}\right]$(15) ) can be derived from the steady-state lateral force balance of a single-track model [14].

10 The wheelbase L and the track width W of the vehicle are assumed to be known.

11 We assume that ${\tau }_{\mathrm{\Omega }}$ is a function of only $v_x$, even if in general it also depends on $a_x$. A second option is to learn ${\tau }_{\mathrm{\Omega }}(v_x,a_x)$ with a neural network. However, a shallow network structure should be adopted, so as to limit the computational complexity when using the neural model for NMPC.

12 In the MPC model we use (14(14) ${\tau }_{\mathrm{\Omega }}\left(v_x\right(t\left)\right)\cdot \dot{\mathrm{\Omega }}\left(t\right)+\mathrm{\Omega }\left(t\right)={\mathrm{\Omega }}_{ss}\left(t\right)$(14) ), but we do not relate ${\mathrm{\Omega }}_{ss}$ with the steering angle δ using (15(15) $\delta \left(t\right)-\frac{{\mathrm{\Omega }}_{ss}\left(t\right)}{v_x\left(t\right)}L=K_{us}\left(a_y\right(t),v_x(t\left)\right)⟶{\mathrm{\Omega }}_{ss}\left(t\right)=v_x\left(t\right)\left[\frac{\delta \left(t\right)-K_{us}\left(a_y\right(t),v_x(t\left)\right)}{L}\right]$(15) ). We develop instead a neural model to compute the steering wheel angle ${\delta }_D$ as a function of $\{\mathrm{\Omega },v_x,a_x\}$.

13 It is assumed that the recorded maximum accelerations are reachable at any time. It might instead happen that the peak values of $\{a_x,a_y\}$ are feasible for only a short amount time (in transient conditions), which may lead to an overestimation of the G-G-v diagram. An improvement of such limitation will be proposed in future work.

14 A video showing the simulation result is available at https://www.youtube.com/watch?v=h5eW01xXWaw.