Time-Optimal Control for High-Order Chain-of-Integrators Systems with Full State Constraints and Arbitrary Terminal States (Extended Version)

TL;DR

A novel notation system and theoretical framework is established, successfully providing the switching manifold for high-order problems in the form of switching law, and a trajectory planning method named the manifold-intercept method (MIM) is developed.

Abstract

Time-optimal control for high-order chain-of-integrators systems with full state constraints and arbitrarily given terminal states remains a challenging problem in the optimal control theory domain, yet to be resolved. To enhance further comprehension of the problem, this paper establishes a novel notation system and theoretical framework, providing the switching manifold for high-order problems in the form of switching laws. Through deriving properties of switching laws regarding signs and dimension, this paper proposes a definite condition for time-optimal control. Guided by the developed theory, a trajectory planning method named the manifold-intercept method (MIM) is developed. The proposed MIM can plan time-optimal jerk-limited trajectories with full state constraints, and can also plan near-optimal non-chattering higher-order trajectories with negligible extra motion time compared to optimal profiles. Numerical results indicate that the proposed MIM outperforms all baselines in computational time, computational accuracy, and trajectory quality by a large gap.

Figures (10)

• Figure 1: A fifth order trajectory planned by the proposed method. $M_5$, $M_4$, $M_3$, $M_2$, $M_1$, and $M_0$ are the upper bounds of position, velocity, acceleration, jerk, snap, and crackle, respectively. The trajectory can be represented as $\underline{0}\overline{0}\left(\overline{3},2\right)\overline{0}\underline{0}\overline{03}\underline{01}\overline{0}\underline{2}\overline{01}\underline{0}\overline{4}\underline{01}\overline{0}\underline{2}\overline{01}\underline{0}\underline{3}\overline{01}\underline{0}\overline{0}$ in Section \ref{['sec:CostateSystemBehaviorAnalysis']}.
• Figure 2: A 3rd order optimal trajectory represented by $\overline{01}\underline{0}\overline{2}\underline{01}\overline{0}$ in this paper, where $\lambda_0>0$. (a) The state vector. (b) The costate vector.
• Figure 3: A 3rd order optimal trajectory represented by $\underline{01}\overline{0}\left(\overline{3},2\right)\overline{0}\underline{0}\overline{0}$ in this paper, where $\lambda_0>0$. (a) The state vector. (b) The costate vector.
• Figure 4: Switching laws for 2 optimal problems with the same terminal state vector ${\boldsymbol{x}}_{\mathrm{f}}$, i.e., ${\mathcal{P}}_1={\mathcal{P}}\left({\boldsymbol{x}}_0^{\left(1\right)},{\boldsymbol{x}}_{\mathrm{f}};{\boldsymbol{M}}\right)$ and ${\mathcal{P}}_2={\mathcal{P}}\left({\boldsymbol{x}}_0^{\left(2\right)},{\boldsymbol{x}}_{\mathrm{f}};{\boldsymbol{M}}\right)$. Among them, $n=2$, ${\boldsymbol{M}}=\left(1,1,1\right)$, $\mathcal{S}\left({\mathcal{P}}_1\right)=\underline{0}\underline{1}\overline{0}$, and $\mathcal{S}\left({\mathcal{P}}_2\right)=\underline{0}\overline{0}$.
• Figure 5: Examples of the optimal-trajectory manifolds. (a) Second order. ${\boldsymbol{M}}=\left(1,1,1.5\right)$, ${{\boldsymbol{x}}_\mathrm{f}}={\boldsymbol{0}}$. (b) Third order. ${\boldsymbol{M}}=\left(1,1,1.5,4\right)$, ${{\boldsymbol{x}}_\mathrm{f}}={\boldsymbol{0}}$. The parameters and the switching surfaces are the same as he2020time.

Theorems & Definitions (46)

• Proposition 1: Bang-Singular-Bang Control Law
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Theorem 1
• proof
• Definition 1
• Definition 2