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Optimal Guidance with Terminal Intercept-Angle Constraints and Acceleration Bounds

Yoav Azaryahu, Vitaly Shaferman

arXiv:2609.28381v1eess.SYcs.SYmath.OC

TL;DR

Terminal-angle constraints can increase required missile acceleration, making explicit acceleration bounds relevant to interception guidance. The paper derives a bounded optimal guidance law using polynomial geometry approximations and evaluates it in nonlinear simulations. When saturation occurs, simulations show improved miss and terminal-angle accuracy over unconstrained counterparts.

  • Problem

    Terminal-angle-constrained guidance can demand increased missile acceleration, motivating guidance that explicitly accounts for acceleration limits.

    Source: §I · p. 2

  • Method

    The paper derives an optimal guidance law with an explicit hard acceleration-command limit and uses polynomial approximations of line-of-sight projection coefficients.

    Source: §VIII · p. 30

  • Results

    The simulations show substantially improved miss distance and terminal-angle accuracy relative to unconstrained guidance when saturation is encountered.

    Source: §VIII · p. 30

  • Takeaways & Limitations

    When saturation occurs near both the beginning and end of an engagement, the bounded law anticipates upcoming saturation and compensates through earlier maneuvers.

    Source: §VIII · pp. 30–31

Abstract

from arXiv · show

Terminal intercept-angle control against a maneuvering target can substantially increase the required missile acceleration, potentially leading to saturation and interception failure unless acceleration limits are explicitly addressed. The engagement is therefore formulated as a linear-quadratic optimal-control problem with bounded acceleration commands. Polynomial approximations of the line-of-sight projection coefficients are used to better represent the nonlinear engagement geometry and estimate the time-to-go. The bounded optimal command is derived over saturated and unsaturated arcs, whose switching times are computed at each guidance step. The guidance law is derived for arbitrary linear missile dynamics and implemented for zero-order missile dynamics. For the zero-order model, the conditions under which the terminal demands can be met are derived in closed form, yielding the minimum and maximum reachable commanded terminal intercept angles. Performance is evaluated in nonlinear simulations. Compared with its unconstrained counterparts, the bounded formulation yields substantially smaller miss distances and terminal-angle errors when saturation is encountered. Unlike corresponding bounded miss-only guidance laws, the proposed law does not reduce to its unconstrained counterpart for minimum-phase missile dynamics because the acceleration command can saturate near the end of challenging engagements. The bounded law anticipates this saturation and compensates through earlier maneuvers.

I. Introduction

The paper addresses the gap between terminal-angle-constrained guidance and explicit hard acceleration limits, which matter because angle requirements can increase demanded acceleration. It derives a bounded optimal law and reports improved miss and terminal-angle accuracy when saturation occurs.

I. Introduction

  • Terminal-angle-constrained guidance can substantially increase required missile acceleration, while finite maneuver authority can cause saturation and degrade miss and terminal-angle accuracy.The introduction identifies a gap: prior optimal-control approaches did not combine a prescribed terminal intercept angle with an explicit constant hard acceleration bound.Source: §I · pp. 1–3
  • The proposed linear-quadratic optimal guidance law imposes a prescribed terminal intercept angle under an explicit constant hard acceleration limit.The law is derived for arbitrary linear missile dynamics and implemented and analyzed for zero-order dynamics.Source: §I · p. 3
  • Time-varying polynomial approximations of line-of-sight projection coefficients improve representation of nonlinear geometry and the corresponding time-to-go estimate.For zero-order missile dynamics, the paper derives the minimum and maximum achievable intercept angles.Source: §I · p. 3
  • Unlike corresponding miss-only bounded laws for minimum-phase missiles, the angle-constrained law can saturate near intercept and anticipates that demand through earlier maneuvers.Its acceleration command generally does not converge to zero near the end of the engagement when a terminal angle is imposed.Source: §I · pp. 3–4
  • When saturation occurs, the bounded law yields substantially smaller miss distances and terminal-angle errors than its unconstrained counterparts.The improvement also holds in scenarios with extended saturation during the engagement.Source: §I · p. 4

II. Model Derivation

The model begins with planar relative engagement geometry and linearizes the nonlinear kinematics about a nominal collision geometry. It then formulates terminal-angle interception as a bounded quadratic-control problem.

A. Engagement Geometry and Nonlinear Relative Kinematics

B. Linearization Model for Guidance Law Derivation

  • A small-deviation linearization represents lateral relative dynamics using missile and target accelerations projected perpendicular to the initial line of sight.Time-varying line-of-sight projection terms are approximated by polynomials in time-to-go; this extends the constant-projection case and supports closed-form influence-function and interval-integral evaluation.Source: B. Linearization Model for Guidance… · p. 5

A. Optimization Problem

  • The optimization minimizes a quadratic cost for interception at a required terminal angle, with non-negative weights and a bounded control.In the limits a→∞ and b→∞, perfect interception and the prescribed intercept angle, respectively, are enforced.Source: A. Optimization Problem · p. 7

B. Coordinate Change and Order Reduction

A terminal-projection coordinate change reduces the guidance problem to projected miss and angle variables. The paper derives unconstrained and bounded controllers, with the bounded solution organized over saturated and unsaturated arcs.

B. Coordinate Change and Order Reduction

  • The terminal-projection transformation reduces the problem order by projecting the state to terminal values under zero missile command and the assumed target maneuver.The projected variables are the zero-effort miss and zero-effort angle, and the reduced dynamics depend only on time-to-go and control.Source: B. Coordinate Change and Order Reduction · pp. 7–8

A. Unconstrained Optimal Controller

  • The unconstrained controller is obtained by substituting the optimal terminal values and projected variables into the optimal command structure.The projected variables include zero-effort miss and zero-effort angle for a constant target acceleration.Source: A. Unconstrained Optimal Controller · p. 9

B. Bounded Optimal Controller

  • The bounded optimum is the pointwise saturation of the unconstrained command structure, constrained by |u| ≤ u_max.The bounded problem's terminal zero-effort miss and angle may differ from those in the unbounded problem.Source: B. Bounded Optimal Controller · pp. 9–10

1. Saturated and unsaturated arcs and terminal-value system

  • Terminal integrals sum the contributions of saturated and unsaturated arcs, allowing their values to be assembled from the corresponding interval integrals.On saturated arcs the command is constant at ±u_max; on unsaturated arcs it follows the bounded unconstrained command, and the terminal values satisfy a linear system.Source: §1 · pp. 10–11

2. Switching times and arc sequence

The switching structure is solved jointly with terminal values because the arc sequence is not known in advance. The paper specializes the law to zero-order dynamics and cubic geometry coefficients to derive closed-form integrals and reachability conditions.

2. Switching times and arc sequence

  • Switching times occur where the unsaturated command reaches ±u_max, and are solved jointly with terminal values because the number and sequence of arcs are unknown.Candidate arc sequences must be checked afterward for consistency with the saturation bounds.Source: §2 · pp. 11–12

V. Zero-Order Missile Dynamics with Cubic Geometry Coefficients

  • For zero-order missile dynamics, the bounded law is specialized with cubic geometry coefficients to analyze when the commanded terminal angle can be reached.This specialization derives conditions for meeting the terminal demands.Source: §V · p. 12

A. Zero-Order Dynamics and Integrals

  • Cubic polynomials approximate the missile and target geometry coefficients, enabling the zero-order model's influence functions and interval integrals to be evaluated.The zero-order model uses G_M(s) = 1.Source: A. Zero-Order Dynamics and Integrals · pp. 12–13

B. Typical Bounded Command Arcs

With a monotonicity assumption on the zero-order geometry coefficient, the bounded command has five typical arc patterns and at most two switches. For fixed angle correction, achievable miss corrections form an interval, enabling closed-form reachability conditions for the terminal demands.

B. Typical Bounded Command Arcs

  • Without the monotonicity assumption, the fourth-order unsaturated command can have up to four roots at each bound and switch up to eight times.The authors say the approximation order can be reduced if the condition fails; the general derivation, algorithm, and simulations do not rely on it.Source: B. Typical Bounded Command Arcs · pp. 13–14
  • When the geometry coefficient is strictly monotonic, the unsaturated command is monotonic and reaches each acceleration bound at most once, yielding at most two switches.The assumption holds when the derivative condition in Eq. (51) is satisfied; the presentation of typical arcs and the reachability analysis use this condition.Source: B. Typical Bounded Command Arcs · pp. 13–14
  • The bounded command has five typical patterns: SUS, US, SU, fully saturated S, and fully unsaturated U.The letters list saturated and unsaturated arcs from the current instant toward intercept; the saturated-arc signs and switching times are determined separately.Source: B. Typical Bounded Command Arcs · p. 14, Figure 2 · p. 14
  • In fully unsaturated mode U, the terminal values reduce to the unconstrained solution.For zero-order dynamics, the geometry coefficients use a cubic approximation, with B̃1(ξ) = −ξK_M(ξ).Source: B. Typical Bounded Command Arcs · pp. 13–14, §1 · p. 15

1. The reachable demands

  • The angle correction alone is achievable if and only if Eq. (55) holds; this condition is independent of R1, and when it fails, no admissible command can achieve the required angle correction.Source: §1 · p. 15
  • At a fixed feasible angle correction, admissible commands produce every miss correction between the interval endpoints.Convex combinations of commands preserving the angle correction remain admissible, and their miss corrections span the values between those of the original commands.Source: §1 · p. 15

2. The extreme commands

For a feasible angle correction, two single-switch bang-bang commands attain the largest and smallest achievable miss corrections. These extrema bound the miss corrections of all admissible commands producing that angle correction.

2. The extreme commands

  • The two extrema are single-switch bang-bang commands whose switching times are fixed by the angle-correction condition.The two commands use opposite signs at their saturated arcs.Source: §2 · pp. 15–16
  • For any feasible angle correction, u+ and u− produce the largest and smallest miss corrections among admissible commands, so their values bound the achievable interval.The terminal demands are feasible when the required miss correction lies between these endpoints, in addition to satisfying the angle-correction condition.Source: §2 · pp. 16–17
  • When the angle-correction condition holds with equality, both extreme commands collapse to the sole admissible constant command u ≡ u_max sgn(R2).The miss-correction interval then degenerates to one value.Source: §2 · p. 17

3. The limiting commanded angles for interception

For zero-order missile dynamics, the required miss correction determines whether a commanded terminal angle is reachable. The reachable commanded angles form an interval bounded by the angles at which the miss-correction interval just contains the required correction.

3. The limiting commanded angles for interception

  • As the commanded angle decreases, both miss-correction endpoints increase, shifting the achievable interval upward while the required miss correction stays fixed.The endpoint derivatives are nonpositive and strictly negative when their corresponding switch times lie in (0, t_go].Source: §3 · p. 18
  • If the required miss correction lies outside the authority available without an angle demand, no admissible command can produce it at any commanded angle.This boundary is expressed by Eq. (66).Source: §3 · p. 18
  • Reachable commanded terminal angles form an interval, with its limits set where the required miss correction meets an endpoint of the achievable miss-correction interval.A commanded angle is reachable only when the required miss correction lies within that interval and the angle correction itself is feasible.Source: §3 · pp. 17, 19
  • For higher-order missile dynamics, the angle correction no longer fixes the extreme-command switch times through the command integral alone.The miss-correction interval structure survives by linearity, but finding its endpoints jointly with the angle correction makes the analysis more complicated.Source: §3 · p. 19

VI. Implementation

The implementation repeatedly updates engagement geometry and time-to-go, estimates the current projected variables, and solves for a bounded command. It initializes the solution with unbounded pre-runs and jointly solves terminal values and switching times for admissible saturated/unsaturated arc sequences.

VI. Implementation

  • At each guidance step, the method updates geometry and time-to-go, evaluates projected variables, and solves for the bounded command.It predicts a trajectory using the current open-loop bounded solution, refits geometry coefficients, then resolves time-to-go and the command.Source: §VI · p. 19, A. Initialization, Geometry… · p. 20

A. Initialization, Geometry Coefficients, and Time-to-Go

  • The first guidance step uses unbounded OGL-CTIA and IAOGL pre-runs to fit cubic geometry coefficients and initialize the bounded solve.The time-to-go is solved from the fitted geometry; a positive real polynomial root nearest the previous estimate is selected, with a kinematic estimate as fallback.Source: A. Initialization, Geometry… · p. 20

B. Approximation of the Current Projected Variables

  • Projected-variable estimation uses seeker-measurable range and line-of-sight rate, avoiding the approximation r ≈ Vc tgo when closing speed varies.The unmeasured quantity y + ẏtgo is approximated using the collision-triangle small-deviation assumption.Source: B. Approximation of the Current… · pp. 20–21

1. The Candidate System and Its Solution

  • Candidate arc sequences are accepted only after convergence and an arcwise admissibility check; the described procedure converged in all tested cases.For zero-order dynamics there are at most two switches; if a candidate fails, the solver tests alternatives, with a fresh unconstrained terminal-value initialization as a rare second fallback.Source: §1 · pp. 21–23
  • For each candidate arc sequence, the solver jointly determines two terminal values and its switching times from 2 + m equations.The command and terminal values are coupled; missile dynamics enter through two terms and their integrals, allowing the procedure to apply to any missile order.Source: §1 · pp. 21–22

2. Mode Transitions

The solver handles changes in saturation structure by replacing the final switching condition with complementarity, allowing an arc-sequence candidate to continue through its boundary.

2. Mode Transitions

  • As time-to-go shrinks, the active arc at the current instant can vanish when its switching time reaches the end of the interval.The m- and (m−1)-switch descriptions coincide at the transition.Source: §2 · p. 23
  • For candidates with at least one switch, a complementarity condition replaces the last switching equation to carry the candidate across an arc boundary.The normalized arc length p and margin q encode whether the final saturated arc is present or absent.Source: §2 · p. 23
  • The Fischer–Burmeister function enforces nonnegative p and q with pq = 0, while remaining smooth away from the origin.A candidate with a zero-length final arc converges to the same solution as the candidate with that arc removed.Source: §2 · pp. 23–24

D. Algorithm Summary

In nonlinear simulations, the bounded law achieves smaller terminal errors than the unconstrained comparison laws when acceleration saturation occurs. Its stepwise algorithm refines geometry and time-to-go before solving for bounded commands, while the simulations show how demanding intercept angles reshape trajectories and acceleration histories.

D. Algorithm Summary

  • The guidance loop initializes with unbounded OGL-CTIA and IAOGL pre-runs, then refines geometry and time-to-go and solves for terminal values, switching times, and the bounded command.The bounded command is applied at each guidance step.Source: D. Algorithm Summary · p. 24

VII. Simulation Results

  • Across the intercept-angle sweep, all cases intercept, while more demanding negative angles require greater lateral displacement and steeper approaches.The negative-angle cases also require earlier, larger commands and longer intervals at the acceleration bound; larger positive angles have milder command profiles.Source: §VII · p. 26

A. Analysis of a Representative Sample Run

  • For the χc = −6° run, IABOGL yields substantially smaller terminal errors than IAOGL and OGL-CTIA despite significant near-end saturation.Table 2 summarizes miss distance and terminal-angle error for this representative case.Source: A. Analysis of a Representative Sample… · p. 27, Table 2 · p. 27
  • In the representative run, IABOGL saturates at the start and enters the final saturated segment earlier than the unbounded laws, anticipating acceleration demands at both ends of the horizon.The saturation sequence is SUS → US → S.Source: A. Analysis of a Representative Sample… · pp. 27–28
  • The polynomial time-to-go prediction remains almost identical to the true value, whereas the kinematic estimate deviates more and degrades terminal accuracy by an order of magnitude when substituted.Despite saturation, IABOGL keeps its predicted miss-related terminal quantity near zero and its predicted angle-related quantity near the commanded angle.Source: A. Analysis of a Representative Sample… · pp. 28–29

B. Parametric Sweep Over the Commanded Terminal Angle

The terminal-angle sweep shows that IABOGL satisfies both criteria over a wider command range than IAOGL and OGL-CTIA. The conclusions attribute its advantage under saturation to accounting for bounded arcs and anticipating later saturation with earlier maneuvers.

B. Parametric Sweep Over the Commanded Terminal Angle

  • IABOGL meets both criteria from approximately −11.5° to +85°, versus +5° to +70° for OGL-CTIA and +2° to +69° for IAOGL.The bounded formulation extends the successful region by about 16° toward negative commands and 15° at the positive end; gains concentrate in cases saturated near both engagement start and end.Source: B. Parametric Sweep Over the Commanded… · p. 29
  • The analytical admissible interval [−13.0°, +83.9°] predicts both observed capture-region boundaries, which lie within 1–2° of the predicted values.The miss distance and terminal-angle error remain small for milder commands and grow as the command approaches either limit.Source: B. Parametric Sweep Over the Commanded… · pp. 29–30
  • The sweep uses 0.5 m miss and 0.5° angle criteria; relaxing either criterion would expand the observed successful region and make the reachability prediction more conservative.The tested command interval is χ_c ∈ [−15°, +90°].Source: B. Parametric Sweep Over the Commanded… · pp. 29–30

VIII. Conclusions

  • The bounded optimal-control problem uses zero-effort miss and angle, with switching times and terminal values obtained by solving the per-arc optimality system.The law includes an explicit hard acceleration-command limit and was evaluated in nonlinear simulation.Source: §VIII · p. 30
  • Polynomial approximations of line-of-sight projection coefficients represent nonlinear geometry and determine time-to-go; purely kinematic estimates degrade miss-distance and terminal-angle performance.The geometry-based time-to-go is an important part of the guidance law.Source: §VIII · p. 30
  • For the zero-order model, the law derives closed-form conditions for terminal-demand reachability, including minimum and maximum reachable commanded intercept angles.The analytical limits agreed well with nonlinear simulation.Source: §VIII · p. 30
  • Across a broad range of intercept angles, IABOGL substantially improves miss distance and terminal-angle accuracy over IAOGL and OGL-CTIA when saturation occurs.The largest improvement occurs with saturation near both engagement start and end, as IABOGL anticipates later saturation through earlier maneuvers.Source: §VIII · pp. 30–31
  • Unlike corresponding bounded miss-only laws for minimum-phase missiles, the intercept-angle-constrained law does not reduce to its unconstrained counterpart because its command generally remains nonzero and saturates near interception.This behavior is typical in challenging scenarios.Source: §VIII · p. 31

Appendix A. Reduced-Order Dynamics and Zero-Effort Variables

The appendices derive reduced-order dynamics and zero-effort variables, then provide Jacobian details for the inner solver. They also specify how switching conditions are handled at mode boundaries.

Appendix A. Reduced-Order Dynamics and Zero-Effort Variables

  • Appendix A derives influence functions and closed-form expressions for the zero-effort variables.The derivation uses time-to-go and the transition matrix, then substitutes polynomial approximations of the line-of-sight projection coefficients.Source: Appendix A · p. 31, A. Reduced-Order Dynamics · pp. 31–32

A. Reduced-Order Dynamics

  • The reduced-order dynamics derivation obtains entries of the transition matrix from the system structure, including φ11 = 1, φ12 = t_go, and φ33 = 1.It also derives the missile- and target-related row blocks.Source: A. Reduced-Order Dynamics · pp. 31–32

B. Zero-Effort Variables

  • The zero-effort-variable derivation combines homogeneous and target-driven components; its target-term simplification assumes constant future target maneuver a_T.The appendix states that this completes the derivation of the zero-effort expressions.Source: B. Zero-Effort Variables · pp. 32–33

Appendix B. Inner-Solver Jacobian

  • Appendix B gives the closed-form Jacobian for the inner-solver system, differentiating terminal-state equations with respect to terminal values and switching times.Switching times enter the terminal equations through arc-integral limits and saturated contributions; the ordinary-condition Jacobian is block lower-triangular.Source: Appendix B · p. 33, A. Terminal States and… · pp. 33–34

Appendix B.B, where the command at 푡푔표푠푚is no longer on the bound.

  • The implementation scales switching rows and corresponding Jacobian rows by 1/u_max, leaving the Newton step unchanged while keeping residual components comparable.At a solution with ordinary switching conditions, the terminal-row switching columns vanish.Source: Appendix B · p. 34, A. Terminal States and… · p. 34

B. Mode-Boundary Row

  • For candidates with at least one switch, the last switching condition is replaced by the Fischer–Burmeister condition at a mode boundary.The replacement row depends on the terminal time and terminal values, and is dimensionless and not rescaled.Source: B. Mode-Boundary Row · pp. 34–35
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