Time-Optimal Switching Surfaces for Triple Integrator under Full Box Constraints

TL;DR

This work tackles time-optimal control for a triple integrator under full box constraints, including asymmetric bounds and active position constraints. It develops a complete geometric characterization of switching surfaces via Pontryagin's Maximum Principle, including tangent markers, and introduces an augmented switching-law (ASL) framework. A pair of complete, Gröbner-basis–enabled algorithms compute time-optimal trajectories in microseconds, outperforming optimization-based baselines and achieving 100% feasibility on tested instances. The results enable robust, real-time trajectory planning under non-stationary terminals and tight state- and input-bounds, marking the first full characterization for this class of problems.

Abstract

Time-optimal control for triple integrator under full box constraints is a fundamental problem in the field of optimal control, which has been widely applied in the industry. However, scenarios involving asymmetric constraints, non-stationary boundary conditions, and active position constraints pose significant challenges. This paper provides a complete characterization of time-optimal switching surfaces for the problem, leading to novel insights into the geometric structure of the optimal control. The active condition of position constraints is derived, which is absent from the literature. An efficient algorithm is proposed, capable of planning time-optimal trajectories under asymmetric full constraints and arbitrary boundary states, with a 100% success rate. Computational time for each trajectory is within approximately 10$μ$s, achieving a 5-order-of-magnitude reduction compared to optimization-based baselines.

Figures (6)

• Figure 1: Illustration of the double integrator. (a) Time-optimal trajectories from various initial states ${\boldsymbol{x}}_0$ to a fixed terminal state ${\boldsymbol{x}}_{\text{f}}$. (b) 2nd-order optimal-trajectory manifolds. (c-d) The state and costate profiles of the time-optimal trajectory between ${\boldsymbol{x}}_0^{(2)}=(0.72,1.182)$ and ${\boldsymbol{x}}_{\text{f}}$. In this example, we assign ${\boldsymbol{x}}_{\text{f}}=(0.5,0.2)$, $-0.9\leq u\leq1$, and $(-0.9,-1)\leq{\boldsymbol{x}}\leq(1,1.5)$.
• Figure 2: An example of the triple integrator without position constraints. (a) Time-optimal trajectories from different initial states ${\boldsymbol{x}}_0$ to a fixed terminal state ${\boldsymbol{x}}_{\text{f}}$. (b) Switching manifolds. (c-d) The state and costate profiles of the time-optimal trajectory between ${\boldsymbol{x}}_0^{(9)}=(-0.6,1.4,1.6)$ and ${\boldsymbol{x}}_{\text{f}}$, where the ASL is $\underline{0}\underline{1}\bar{0}\underline{2}\bar{0}\bar{1}\underline{0}$. In this example, we assign ${\boldsymbol{x}}_{\text{f}}=(0.5,0.2,0.1)$, $-0.9\leq u\leq1$, and $(-0.9,-1,-\infty)\leq{\boldsymbol{x}}\leq(1,1.5,\infty)$.
• Figure 3: Partition of the state space ${\mathbb{R}}^3$ by ASLs. The terminal state and constraints are the same as those in Fig. \ref{['fig:3order_manifold_inf_position']}.
• Figure 4: Switching manifolds of the triple integrator with full constraints, where $(-0.9,-1,0)\leq{\boldsymbol{x}}\leq(1,1.5,2.4)$ and $-0.9\leq u\leq1$. (a) ${\boldsymbol{x}}_{\text{f}}=(0.5,0.2,0.2)$. (b) ${\boldsymbol{x}}_{\text{f}}=(0.5,0.3,0.075)$. In (a1) and (b1), switching manifolds consider the position constraints. In (a2) and (b2), switching manifolds do not consider position the position constraints. For each case, two trajectories, i.e., ${\boldsymbol{x}}^{(1)}(t)$ and ${\boldsymbol{x}}^{(2)}(t)$, are planned under the same boundary conditions but without and with considering position constraints, respectively. The initial states in (a2) and (b2) are ${\boldsymbol{x}}_0=(0.5,-0.6292,0.2636)$ and $(0.98,-0.5944,0.2722)$, respectively.
• Figure 5: State and costate profiles of time-optimal trajectories with the tangent marker $(\underline{3},2)$. In this figure, (a) and (b) correspond to ${\boldsymbol{x}}^{(2)}(t)$ Fig. \ref{['fig:3order_manifold_constrained_position']}(a) and (b) with ASLs $\bar{0}\bar{1}\underline{0}(\underline{3},2)\underline{0}\underline{1}\bar{0}\underline{0}$ and $\bar{0}\bar{1}\underline{0}\bar{0}(\underline{3},2)\bar{0}\underline{0}$, respectively.

Theorems & Definitions (9)

• Remark 1
• Remark 2
• Theorem 1
• proof
• Remark 3
• Theorem 2
• proof
• Remark 4
• Remark 5