Primer Vector

Author: Tianjiang Shuo

Website: https://cislunarspace.cn

Definition

The primer vector is a term coined by Lawden (1963) in his foundational work Optimal Trajectories for Space Navigation. It is defined as the negative of the velocity costate:

It is the central adjoint quantity produced by Pontryagin's Minimum Principle for optimal spacecraft control: it fixes the optimal thrust direction and throttle-switching instants in the continuous-thrust case, and the impulse times, directions, and the question of whether to add further impulses in the impulsive case. Lawden explained the name in a 1990 letter to Prussing: he served in the artillery during World War II, where a primer charge initiates the burning of cordite; analogously, "$p=1$" is the signal for the rocket motor to ignite (Prussing 2010).

Derivation: from the Minimum Principle to the primer vector

Consider a variable-mass spacecraft in a central gravity field,

with thrust direction $\boldsymbol{\alpha}$ ($\|\boldsymbol{\alpha}\|=1$), throttle $u\in[0,1]$, and exhaust velocity $c=I_{sp}g_0$. The Hamiltonian is

Minimizing over $\boldsymbol{\alpha}$ yields the optimal thrust direction

i.e., the optimal thrust direction is along the primer vector. Substituting back gives the control-coupling term $H_c=u\,\Phi(t)$ with the switching function

The Minimum Principle then gives $u^{*}=0$ when $\Phi>0$ (coast) and $u^{*}=1$ when $\Phi<0$ (full thrust) — the structure underlying Bang-bang control.

The primer vector equation

The costate equations $\dot{\boldsymbol{\lambda}}_v=-\boldsymbol{\lambda}_r$ and $\dot{\boldsymbol{\lambda}}_r=-G(\mathbf{r})\boldsymbol{\lambda}_v$, with $G(\mathbf{r})=\partial\mathbf{g}/\partial\mathbf{r}$ the $3\times 3$ gravity-gradient matrix, combine to give the primer vector equation

It is a second-order linear ODE along the trajectory (Lawden 1963, Ch. 3; Prussing 2010, Eq. 2.19) and shares its state transition matrix with the orbital variational equation, so it can be propagated together with the state transition matrix. In the two-body problem $G(\mathbf{r})=\mu(3\hat{\mathbf{r}}\hat{\mathbf{r}}^{\mathrm{T}}-I)/r^3$, where analytical foundations are available (Prussing 1993).

Continuous-thrust necessary conditions

For a constant-specific-impulse (CSI) optimal control problem, the primer vector satisfies (Lawden 1963; Conway 2010, Ch. 2):

1. $\mathbf{p}(t)$ and $\dot{\mathbf{p}}(t)$ are everywhere continuous;
2. The optimal thrust direction is $\boldsymbol{\alpha}^{*}=\mathbf{p}/\|\mathbf{p}\|$;
3. The throttle is decided by the sign of the switching function $\Phi$;
4. At a switching instant, $\Phi=0$, i.e. $\|\mathbf{p}\|=-m\lambda_m/c$.

Lawden accordingly classified thrust arcs into maximum-thrust (MT), null-thrust (NT), and intermediate-thrust (IT, i.e. the singular arc). See Bang-bang Control & Lawden's Arc Law.

Impulsive case: Lawden–Lion–Handelsman conditions

In the high-thrust limit, MT arcs shrink to instantaneous impulses. Lawden (1963) first wrote down the first-order necessary conditions; Lion and Handelsman (1968) put them in the engineering form:

1. $\mathbf{p}(t)$ and $\dot{\mathbf{p}}(t)$ are everywhere continuous;
2. $\|\mathbf{p}(t)\|\le 1$ for all $t$, and impulses can occur only at instants where $\|\mathbf{p}\|=1$;
3. At an impulse, $\mathbf{p}$ is a unit vector in the optimal impulse direction;
4. At an intermediate impulse (not initial or final), $\dot{\mathbf{p}}=\dot{\mathbf{p}}^{\mathrm{T}}\mathbf{p}/\|\mathbf{p}\|=0$.

For linear systems these conditions are also sufficient and bound the number of optimal impulses (Prussing 1993).

Lion–Handelsman gradient method

In practice, given a fixed transfer time and boundary conditions, one usually starts from a non-optimal $N$-impulse solution (e.g., a two-impulse Lambert solution). Lion and Handelsman (1968) derived the cost gradients with respect to three corrective operations:

• Terminal coast: shifting the first/last impulse time — gradient $\partial J/\partial t_i=\pm\|\dot{\mathbf{p}}(t_i)\|$;
• Midcourse impulse: adding an impulse on a sub-arc where $\|\mathbf{p}\|>1$ — gradient $\partial J/\partial\Delta\mathbf{v}=(\|\mathbf{p}\|-1)\,\hat{\mathbf{p}}$;
• Impulse time iteration: nudging impulse times by the residual of $\dot{\mathbf{p}}$.

Jezewski and Rozendaal (1968) embedded these gradients in a nonlinear-programming framework (see Indirect Methods), yielding an algorithm that automatically decides when to add impulses and when to introduce coasts. It remains the standard tool for verifying and improving impulsive-transfer optimality.

Relation to the adjoint-control transformation

Because $u^{*}=\mathrm{sign}(-\Phi)$ with $\Phi$ expressible through $\mathbf{p}$ and $\lambda_m$ alone, the optimal control can be parameterized entirely by $(\mathbf{p},\lambda_m)$. This observation underlies the adjoint-control transformation and costate normalization, which reduce the dimension of the indirect-method search space (Taheri et al. 2016; see Co-state Variables).

Application notes

• Impulsive-transfer optimality test: if $\|\mathbf{p}(t)\|$ exceeds 1 along a two-impulse Lambert solution, an additional impulse or a coast is needed. This is the standard criterion for automating three-impulse cislunar transfer design.
• Continuous-thrust direction command: when simplified tangent-thrust laws fail, taking $\boldsymbol{\alpha}=\mathbf{p}/\|\mathbf{p}\|$ recovers the first-order optimal direction; the remaining freedom is only the throttle schedule, dramatically shrinking the indirect parameter space.
• Multi-body extension: in CR3BP $G(\mathbf{r})$ becomes the rotating-frame Jacobian of the synodic equations, but the primer vector equation retains its form, so the same toolbox serves Earth-Moon $L_1$/$L_2$ transfer analysis.
• Homotopy initialization: the energy-optimal (continuous-thrust) solution gives a closed-form approximation to $\mathbf{p}(t)$, an excellent starting point for homotopy methods that continue toward fuel-optimal bang-bang solutions.

Related concepts

• Bang-bang Control — the throttle structure derived from primer vector magnitude via the switching function
• Co-state Variables — the costate origin of the primer vector
• Pontryagin's Minimum Principle — the mathematical foundation
• Homotopy Method — the numerical workhorse for the primer vector BVP
• Fuel-optimal Control — the dominant cost type for which the primer vector is applied
• Adjoint-Control Transformation — replacing the full costate by $(\mathbf{p},\lambda_m)$
• Indirect Methods — the multiple-shooting NLP framework implementing Lion-Handelsman gradients
• Circular Restricted Three-Body Problem (CR3BP) — the dynamical setting for cislunar primer vector applications

References

• Lawden, D. F. 1963. Optimal Trajectories for Space Navigation. Butterworths, London. (Original definition of the primer vector; continuous and impulsive necessary conditions; three-arc classification.)
• Lion, P. M., and Handelsman, M. 1968. "Primer Vector on Fixed-Time Impulsive Trajectories." AIAA Journal 6(1): 127–132. (Cost gradients for terminal coasts and midcourse impulses.)
• Jezewski, D. J., and Rozendaal, H. L. 1968. "An Efficient Method for Calculating Optimal Free-Space N-Impulse Trajectories." AIAA Journal 6(11): 2160–2165. (NLP implementation of Lion-Handelsman gradients with automatic impulse insertion.)
• Prussing, J. E. 1993. "Equation for Optimal Power-Limited Spacecraft Trajectories." JGCD 16(6).
• Prussing, J. E. 2010. Primer Vector Theory and Applications. In Conway (ed.), Spacecraft Trajectory Optimization, Ch. 2, Cambridge Univ. Press.
• Conway, B. A. (ed.) 2010. Spacecraft Trajectory Optimization. Cambridge Univ. Press.
• Bryson, A. E., and Ho, Y.-C. 1975. Applied Optimal Control. Hemisphere.
• Taheri, E., Kolmanovsky, I., and Atkins, E. 2016. "Enhanced Smoothing Technique for Indirect Optimization of Minimum-Fuel Low-Thrust Trajectories." JGCD 39(11): 2500–2511.