Nonlinear Model Predictive Path Following Controller with Obstacle Avoidance

Abstract

In the control systems community, path-following refers to the problem of tracking an output reference curve. This work presents a novel model predictive path-following control formulation for nonlinear systems with constraints, extended with an obstacle avoidance strategy. The method proposed in this work simultaneously provides an optimizing solution for both, path-following and obstacle avoidance tasks in a single optimization problem, using Nonlinear Model Predictive Control (NMPC). The main idea consists in extending the existing NMPC controllers by the introduction of an additional auxiliary trajectory that maintains the feasibility of the successive optimization problems even when the reference curve is unfeasible, possibly discontinuous, relaxing assumptions required in previous works. The obstacle avoidance is fulfilled by introducing additional terms in the value functional, rather than imposing state space constraints, with the aim of maintaining the convexity of the state and output spaces. Simulations results considering an autonomous vehicle subject to input and state constraints are carried out to illustrate the performance of the proposed control strategy.

Appendix

Using the same arguments as in [20] the following lemma is introduced.

Lemma 1

Consider that Assumption 3 and 4 hold. Consider also a reference path x_ref and assume that for a given state x the optimal solution of ${P^{a}_{N}}(\cdot )$ is such that $x = \bar {x}_{a}(x,x_{\text {ref}})$. Then the function W(x,x_ref) = 0.

Sketch of Proof.

The key idea of the proof is to show that if the system converges to an equilibrium trajectory $\bar {x}_{a}$, then this trajectory is already equal to the reference path x_ref. In other words, the convergence of the system trajectory to the artificial one, $\bar {x}_{a}$, and the convergence of the artificial trajectory to the reference, x_ref, are not consecutive, but simultaneous, and the convergence rate of the former is an upper bound for the latter. The formal proof can be done by contradiction, by assuming that the system state and input, (x,u), converge to an artificial trajectory that is an equilibrium different from the one corresponding to the reference trajectory, i.e., $(x, u) = (\bar {x}_{a},\bar {u}_{a}) \neq (x_{\text {ref}},u_{\text {ref}})$. Then, given that both trajectories $(\bar {x}_{a},\bar {u}_{a})$ and (x_ref,u_ref) belong to a convex set, there exists a feasible trajectory $(\hat {x},\hat {u}) = \beta (\bar {x}_{a},\bar {u}_{a}) + (1-\beta ) (x_{\text {ref}},u_{\text {ref}})$, with β ∈ (0, 1), which produces a cost function smaller than one obtained at $(\bar {x}_{a},\bar {u}_{a})$. This proves that there exists a β ∈ (0, 1) such that the cost of moving the system from $\bar {x}_{a}$ to $\hat {x}$ is smaller than the cost of remaining in the error equilibrium solution $\bar {x}_{a}$. This contradicts the optimality of the solution to problem ${P^{a}_{N}}(\cdot )$, and hence $x=\bar {x}_{a}=x_{\text {ref}}$.

For more details of the complete proof of this Lemma the reader can refer to [20].

Remark 8

Lemma 1 covers the case of a feasible trajectory reference x_ref. However, the analysis made in the proof still holds true for infeasible references. Indeed, by a simple convex analysis, if x_ref is not reachable the system will converge to the optimal feasible equilibrium trajectory with respect to x_ref. This means that the controller finds by itself an optimal trajectory (in the sense of the proximity to x_ref) even in the case x_ref is infeasible/unreachable.