Lambert Guidance Routine

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

The Lambert Guidance Routine (LGR; also called Lambert guidance) is a real-time closed-loop guidance algorithm for the powered-flight phase that embeds the solution of Lambert's problem. Each guidance cycle executes three steps (Burns & Scherock 2004):

Solve Lambert's problem with the current state $(\vec r_M,\vec v_M)$ , the target point $\vec r_2$ , and the remaining time of flight $t_f$ to obtain the "velocity to be had" $\vec v_{\text{LAMBERT}}$ ;
Compute the velocity-to-be-gained $\vec v_G=\vec v_{\text{LAMBERT}}-\vec v_M$ ;
Command thrust along $\vec v_G$ ; when $|\vec v_G|$ falls below a threshold, cut off and coast ballistically to the target.

This is a closed-loop iterative process: every cycle re-solves Lambert from the actual current state, giving natural robustness to disturbances, winds, and thrust deviations without any pre-designed open-loop reference trajectory.

Lambert Computational Plane and Coordinate Conversion

$\vec r_1$ (current vehicle position) and $\vec r_2$ (target) define a unique plane, the Lambert computational plane. The two-dimensional Lambert equation is solved in this plane to give $\vec v_{\text{LAMBERT}}^{(2D)}$ , which is then mapped back to the ECI inertial frame by two rotations about $Z$ and $Y'$ followed by a roll about $X''$ (Burns & Scherock 2004, Appendix). The "solve in 2D, map to 3D" pattern avoids direct 3D iteration.

The transfer angle $\theta_f$ follows from the dot product:

\cos\theta_f=\frac{\vec r_1\cdot\vec r_2}{|\vec r_1|\,|\vec r_2|}

The dimensionless energy parameter $\lambda=|\vec r_1|V^2/\mu$ (twice the ratio of kinetic to potential energy) classifies the trajectory: $\lambda<2$ elliptic, $\lambda=2$ parabolic, $\lambda>2$ hyperbolic. Common implementations iterate the Lambert equation by the secant method, with $\lambda$ restricted to elliptic values.

Distinction from Open-Loop Lambert Transfer and Other Guidance Schemes

vs. open-loop Lambert transfer ("Lambert orbit manoeuvre"): the open-loop use solves Lambert once offline to obtain an initial $\Delta\vec v$ , applies it, and coasts without real-time correction; LGR re-solves every cycle and naturally corrects deviations. The former is a trajectory-design tool, the latter is a guidance law.
vs. Q-guidance and explicit guidance (Explicit Guidance Law): Q-guidance uses costate-gain matrices; explicit guidance integrates the nominal equations and enforces terminal constraints. LGR is characterised by the explicit call to a Lambert solver as the module computing the "velocity to be had".
vs. Apollo LM P64 guidance: P64 is polynomial / Lambda guidance, not Lambert guidance. The occasional Chinese-literature claim that "Lambert guidance was one of the Apollo Guidance Computer's heaviest tasks" is a confusion — Apollo used Lambert solutions for mission planning (on the ground), not for real-time flight guidance.
vs. midcourse Lambert correction: Lambert solutions are also used to compute reference impulses for midcourse corrections (see multi-impulse manoeuvres); that is offline impulse design, distinct from LGR.

Key Engineering Parameters

Scheduled arrival time $T_A$ / time of flight $t_f$ : determines the trajectory shape (high loft vs. low loft) and the terminal velocity-matching $\Delta v$ . Longer $t_f$ implies a higher loft and lower terminal velocity, hence a larger velocity deficit to make up; Powell's method is commonly used to optimise $t_f$ (equivalently, launch delay $T_D$ ) for minimum terminal $\Delta v$ (Burns & Scherock 2004).
Cutoff threshold: shut down main thrust once $|\vec v_G|$ falls below threshold. Too low — tight cycles, demanding real-time performance; too high — large residuals.
Atmospheric pitch program: in the low-atmosphere segment, a fixed fly-out flight-path-angle schedule suppresses drag losses; the loop switches to pure Lambert commands above the atmosphere.
Target position offset (White offset): compensates the systematic bias from Lambert's "instantaneous burn, uniform gravity" assumptions by deliberately biasing the guidance target by a small amount so that the true trajectory lands at the intended point.

Extension: Position and Velocity Matching

Classical LGR only matches terminal position. Burns-Scherock (2004), in an interceptor scenario, add a short fourth-stage impulse so that terminal velocity also matches the target — once both position and velocity match, the interceptor follows the target's trajectory. Procedure:

First pass: guide with LGR to the target position $\vec R_T$ ; estimate $\Delta\vec V$ at closest approach;
Bias the guidance target to $\vec R_{\text{offset}}=\vec R_T-\Delta\vec R$ ( $\Delta\vec R$ is the position deviation induced by the fourth-stage impulse);
The fourth stage ignites along $\Delta\vec V$ at time $\Delta T$ before the target;
Because the second-pass path differs slightly, $\Delta\vec V$ must be iterated to convergence — typically within 2-3 iterations.

Application Notes

Boost-phase intercept guidance: standard mode for ballistic-missile interceptors and strategic target vehicles.
Launch vehicle insertion: boost guidance that arrives at the insertion point at a specified time.
Ascent intercept/rendezvous: real-time guidance from ground/sub-orbital launch to a designated point in space.
No direct control of terminal velocity: classical LGR controls position only; velocity matching requires an additional impulse.

References

Burns and Scherock, 2004, Lambert Guidance Routine Designed to Match Position and Velocity of Ballistic Target (LGR principles, 3D extension, and the fourth-stage position-plus-velocity matching scheme).
Joonhyung Park et al., 2000, Miss Analysis in Lambert Interceptions with Application to a New Guidance Law (miss-distance analysis of Lambert interceptions).