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

Technical Summary: Time-Optimal Switching Surfaces for Triple Integrator Under Full Box Constraints

Problem Statement
This paper addresses the time-optimal control problem for a triple integrator system subject to full box constraints. The system dynamics involve jerk-limited control inputs ($u$) and state constraints on position ($x_1$), velocity ($x_2$), and acceleration ($x_3$). The objective is to transfer the system from an arbitrary initial state $\mathbf{x}_0$ to a fixed terminal state $\mathbf{x}_f$ in minimum time $t_f$.

While time-optimal control for double integrators is well-understood, the triple integrator case presents significant challenges, particularly when dealing with:

1. Asymmetric constraints: Where upper and lower bounds for jerk and states are not equal in magnitude.
2. Non-stationary boundary conditions: Arbitrary initial and terminal states.
3. Active position constraints: Scenarios where the trajectory must touch or slide along the position constraint boundaries ($x_3 = \bar{x}_3$ or $x_3 = \underline{x}_3$).

Existing solutions, such as the commercial Ruckig algorithm (community version), often fail under asymmetric constraints or active position constraints. Previous analytical approaches have either ignored state constraints, assumed symmetric constraints, or failed to provide a complete characterization of the switching surfaces and the active conditions for position constraints.

Methodology
The authors utilize Pontryagin's Maximum Principle (PMP) to characterize the optimal control law, which follows a Bang-Bang-Singular (BBS) structure. The core of the methodology involves a geometric analysis of the state space to derive time-optimal switching surfaces.

1. Augmented Switching Law (ASL): The optimal trajectory is represented as a sequence of arcs (unconstrained bang-bang arcs and constrained arcs where states are fixed) and "tangent markers." A tangent marker occurs when the position constraint becomes active tangentially (i.e., $x_3 = \bar{x}_3$, $x_2 = 0$, and $x_1 \geq 0$).
2. Switching Surface Construction:
   • Without Position Constraints: The authors construct switching manifolds by backward integration from the terminal state $\mathbf{x}_f$. They define surfaces corresponding to specific ASLs (e.g., $\bar{0}\bar{0}\bar{0}$, $\bar{0}\bar{1}\bar{0}\bar{0}$) that partition the 3D state space.
   • With Position Constraints: The paper derives the specific conditions under which position constraints become active. It introduces new switching surfaces (e.g., $\bar{0}(\bar{3}, 2)\bar{0}\bar{0}\bar{0}$) that account for trajectories tangent to the position boundary. Theorem 2 provides a complete classification of all possible ASL forms for the triple integrator with full box constraints, establishing that optimal trajectories consist of sequences of unconstrained arcs and tangent markers.
3. Algorithmic Implementation:
   • Algorithm 1: Determines the relative direction between $\mathbf{x}_0$ and $\mathbf{x}_f$ by analyzing 2nd-order projections, handling cases where switching surfaces are multi-valued.
   • Algorithm 2: Solves the problem without position constraints by connecting states via 2nd-order optimal controls and solving polynomial equations for arc durations.
   • Algorithm 3: Extends the solution to full box constraints. If the unconstrained solution violates position constraints, the algorithm recursively connects the state to a tangent marker (derived from Theorem 2) and solves the sub-problems.
   • The system of equations for arc durations is solved using Gröbner bases, reducing the problem to a single polynomial equation of degree $\leq 6$.

Key Contributions

• Complete Characterization of Switching Surfaces: For the first time, the paper provides a complete analytical characterization of time-optimal switching surfaces for the triple integrator under full box constraints, including the geometric structure of solutions with active position constraints.
• Derivation of Active Conditions: The paper explicitly derives the conditions under which position constraints become active (the "tangent marker" condition), a gap previously absent in the literature.
• Theorem 2 (ASL Classification): It establishes the complete set of allowable Augmented Switching Laws for this problem, proving that optimal trajectories follow specific sequences involving tangent markers.
• Efficient Algorithm: An algorithm is proposed that plans time-optimal trajectories for arbitrary boundary states and asymmetric constraints.

Results
The proposed method was evaluated against three baselines: Ruckig (community version), CasADi (IPOPT-based), and SCP (Sequential Convex Programming).

• Success Rate: The proposed method achieved a 100% success rate in finding feasible solutions for both symmetric (Case 1) and asymmetric with active position constraints (Case 2). In contrast, Ruckig failed in Case 2 due to asymmetry, and SCP/CasADi showed significantly lower success rates (34.1% and 97.7% respectively in Case 2).
• Computational Efficiency: The average computational time per trajectory was approximately 10 $\mu$s (specifically $2.38 \times 10^{-6}$ s for Case 1 and $1.25 \times 10^{-5}$ s for Case 2). This represents a 5-order-of-magnitude reduction compared to optimization-based baselines (CasADi: ~0.28s; SCP: ~7.28s).
• Optimality: The method consistently found globally optimal solutions. In test cases where optimization-based methods converged to local optima (suboptimal ASLs), the proposed method reduced terminal time by up to 90.9% (Case 1) and 73.4% (Case 2).

Significance
The authors claim this work provides the first complete switching surface characterization and an efficient algorithm for the 3rd-order time-optimal control problem under non-stationary terminal conditions and active position constraints. By moving away from iterative optimization toward a geometric, analytical solution, the paper demonstrates that high-precision, time-optimal control can be achieved with microsecond-level computation times, making it highly suitable for real-time industrial applications such as robotic motion and numerical control machining where constraints are often asymmetric and active.