Phase-Based Trajectory Optimization

• Phase-based trajectory optimization is a method that partitions optimal control problems into distinct phases, integrating continuous dynamics with discrete transitions through advanced smoothing techniques.
• It employs both indirect methods (e.g., Pontryagin Maximum Principle) and direct discretization approaches to handle complex phase transition logic effectively.
• The approach is applied in areas like Mars EDL and legged robotics, overcoming computational challenges by smoothing Boolean guard functions into differentiable forms.

Phase-based trajectory optimization refers to a class of methods for solving optimal control problems involving dynamical systems whose evolution is naturally partitioned into sequential segments ("phases"), each governed by distinct dynamics, constraints, and logical transition conditions. These problems are fundamental in aerospace mission design, legged robotics, autonomous vehicles, and hybrid systems, capturing both the continuous and discrete aspects intrinsic to multi-mode operation.

1. Formal Structure of Phase-Based Trajectory Optimization

The canonical mathematical formulation considers $m$ phases indexed by $k=1,2,\ldots,m$. Each phase is associated with a state vector $X(t)\in\mathbb{R}^n$, control $U(t)\in\mathbb{R}^m$, and time interval $[t_{k-1},t_k]$:

• Phase $k$ dynamics: $\dot{X} = f_k(X, U, t)$
• Phase-specific running cost (Lagrangian): $\mathcal{L}_k(X, U, t)$

The objective is a composite cost functional:

$$J = \Phi(X(t_f), t_f) + \sum_{k=1}^m \int_{t_{k-1}}^{t_k} \mathcal{L}_k(X, U, t)\,dt$$

subject to:

• Prescribed initial/terminal conditions, $\Psi_0(X(t_0), t_0)=0$, $k=1,2,\ldots,m$0
• Continuity at phase transitions: $k=1,2,\ldots,m$1
• Transition guards: Boolean conditions, $k=1,2,\ldots,m$2, on the state, time, or control determining if/when the system advances to the next phase

A critical challenge is the treatment of the discrete event logic (often complex combinations of AND, OR, and NOT predicates), which generates a "hybrid" optimal control problem.

2. Representation and Smoothing of Phase Logic

Transition conditions are represented as Boolean guard functions:

• Classical approaches employ explicit case-splits or combinatorial scenario enumeration, which is prohibitive for problems with rich event logic.

The Generalized Relaxed Autonomously Switched Hybrid System (GRASHS) framework (Saranathan, 2024) provides a systematic smoothing methodology for embedding arbitrary discrete logic directly into the continuous optimization, enabling automated treatment of any guard structure:

• Each Boolean phase-guard $k=1,2,\ldots,m$3 (with arbitrary AND/OR/NOT) is converted into Disjunctive Normal Form (DNF):

$k=1,2,\ldots,m$4

• Each predicate $k=1,2,\ldots,m$5 is approximated by a "flipped" sigmoid $k=1,2,\ldots,m$6
• Minterms (AND-clauses) are smoothed as products of sigmoids
• OR-combination (DNF minterms) is smoothed via a hyperbolic tangent:

$k=1,2,\ldots,m$7

• The switching function $k=1,2,\ldots,m$8 acts as a smooth indicator for when phase $k=1,2,\ldots,m$9 is "active," permitting embedding of all logic into the continuous optimal-control problem

This approach removes the need for explicit scenario enumeration and enables robust handling of phase transitions in indirect methods by converting the "multi-point" BVP into a two-point BVP.

3. Indirect and Direct Solution Methodologies

Phase-based trajectory optimization admits solution via indirect and direct methods:

• Indirect methods (e.g., Pontryagin Maximum Principle) require derivation of Euler-Lagrange/Hamiltonian boundary-value problems (BVPs) with co-states, including necessary optimality conditions and transversality at phase junctions. In the smoothed GRASHS approach, the embedding of all switching logic into the dynamics and cost produces a two-point BVP without explicit jump or transversality conditions at the internal phase boundaries (Saranathan, 2024).
• Direct methods discretize state, control, and transition times, often adopting either:
  • Multiplicity-parameterization (multi-point BVP) for fixed phase order and known event timings (Hybrid Trajectory Optimization, HTO) (Turski et al., 2023)
  • Contact-implicit or complementarity-based relaxations for problems with unknown or variable transition structure (CIO and related methods) (Turski et al., 2023)
• Homotopy in the smoothing parameters $X(t)\in\mathbb{R}^n$0 allows modern nonlinear program solvers (shooting, collocation) to converge reliably from a "fully smoothed" to "nearly discontinuous" regime.

4. Treatment of Discrete and Hybrid Events

Trajectory optimization problems span a variety of hybrid event types:

• Mode-dependent continuous dynamics (e.g., powered flight, gliding, parachute descent)
• Mode-dependent path costs
• Discrete state resets (e.g., momentum jumps, mass drops at staging, set-valued maps)
• Logical transition conditions (arbitrary Boolean logic in state, control, or clock)

Phase transition strategies:

• RASHS and older approaches: Only support AND-type guards, requiring scenario splitting for OR logic (Saranathan, 2024)
• GRASHS: Handles arbitrary logic without case splits through DNF+sigmoid+tanh relaxation (Saranathan, 2024)
• Robotics: Contact mode transitions (stance, swing) handled via manual schedule (HTO) or complementarity (CIO), with staged approaches (SCO) combining mode discovery and robust solution (Turski et al., 2023)
• Aircraft: Optimal structure analysis shows concatenations of bang, singular, and boundary arcs reflecting the sequence of constraint (CAS/Mach) and unconstrained arcs (Cots et al., 2022)

5. Representative Applications and Case Studies

Aerospace Mission Phasing

• Mars entry-descent-landing (EDL) with complex phase trigger logic involving nested AND/OR conditions (e.g., parachute deployment triggered by $X(t)\in\mathbb{R}^n$1). GRASHS formulation enables solution without enumerating all orderings of trigger events and automatically recovers correct switching (Saranathan, 2024).

Legged Robotics

• Biped and quadruped motion with contacts. HTO fixes contact sequence, CIO discovers contacts via complementarity, while Staged Contact Optimization (SCO) leverages relaxed CIO to discover contact schedule and then refines using HTO (Turski et al., 2023). SCO achieves solutions an order of magnitude faster than CIO, and with less trial-and-error than HTO.

Method	Handles Arbitrary Logic	Case Splitting Needed	Typical Application Domains
RASHS	No (AND only)	Yes	Aerospace, simple hybrid systems
GRASHS	Yes	No	Aerospace, robotics, general
HTO	N/A (fixed sequence)	Yes	Robotics (user-defined gaits)
CIO	Yes (implicit)	No	Robotics (unknown contact)
SCO	Yes (by staging)	No	Advanced legged robot behaviors

6. Algorithmic Strategies and Solver Guidelines

A unified solution strategy for phase-based problems involving arbitrary phase logic, as in GRASHS (Saranathan, 2024), is as follows:

1. Convert each phase's trigger-guard to DNF boolean expression.
2. Smooth each predicate to a differentiable sigmoid.
3. Product over ANDs (minterms), sum for ORs, then apply tanh smoothing to combine into a phase activation function.
4. Replace original multi-modal dynamics and cost with a sum over all phases, each modulated by their activation function.
5. Initialize with low smoothing (small $X(t)\in\mathbb{R}^n$2) to aid convergence, solve with indirect (shooting or collocation) solvers.
6. Gradually increase smoothing parameters using a homotopy or continuation approach.
7. Final solution yields continuous phase trajectories, with active segments corresponding to where $X(t)\in\mathbb{R}^n$3 switches.

Key practical guidelines:

• Transitions are detected when switching functions cross 0.5.
• Smaller $X(t)\in\mathbb{R}^n$4 (more smoothing): easier convergence, less phase boundary accuracy.
• Larger $X(t)\in\mathbb{R}^n$5 (less smoothing): higher fidelity, but solver convergence more demanding.
• Smoothing bridges the gap between fully relaxed/complementarity-based and classical sharp-contact/hard-switching NLP strategies.

7. Extensions and Comparative Perspectives

Current phase-based trajectory optimization research encompasses several lines:

• Generalization to arbitrary hybrid systems with nested or sequential mode transitions, via logic-to-DNF reduction and smooth embedding (Saranathan, 2024).
• Dynamic scheduling of modes/contact sequences by hybrid reinforcement learning, as in the transformer-based policy for multi-phase spacecraft trajectories (Jain et al., 14 Nov 2025).
• Formulation of multiphase aircraft climb with concatenated bang, singular, and boundary arcs; practical procedures (e.g., constant CAS/Mach) are shown to be near optimal but slightly suboptimal compared to fully optimal patchwise controls (Cots et al., 2022).
• Comparative studies reveal that smoothing-based approaches eliminate most of the computational complexity (case split, combinatorial explosion) and enable solution of complex problems previously intractable by purely discrete or direct methods.

A plausible implication is that phase-based trajectory optimization grounded in logic-to-DNF smoothing and embedded phase activation functions (as in GRASHS) provides a universal and scalable framework for hybrid systems, circumventing the scenario-proliferation that historically limited the robustness and flexibility of indirect or direct approaches in high-dimensional or logic-heavy multi-phase settings (Saranathan, 2024, Turski et al., 2023, Cots et al., 2022).