The Minimum-Time Thrust-Vector Control Law in the Apollo Lunar-Module Autopilot

Abstract

The descent-propulsion-system engine of the Apollo lunar module is mounted in gimbals with actuators so that the direction of the thrust vector can be adjusted with respect to the spacecraft center-of-mass. A single slow actuator drive rate of 0.2 deg/sec is available. For spacecraft attitude control, a minimum-time control law was developed. This control law - termed the optimal control law-is third order. The state variables being controlled are the spacecraft attitude, angular velocity and angular acceleration. The spacecraft angular acceleration is proportional to the first integral of the control variable, which is a command to drive the engine gimbal at its 0. 2 deg/sec drive rate. The explicit feedback formulation of the optimal control law was derived by satisfying the necessary conditions of Pontryagin, using the techniques provided by Athans and Falb. This control law has been implemented in the digital computer of the onboard primary navigation, guidance, and control system. Early simulation results indicated that lags associated with reversing the gimbal drive command plus sampled-data effects contributed to undesirable overshoot in the closed-loop transient response and unaceptably large limit-cycle amplitudes in the steady-state response. This difficulty was overcome simply by reducing the gain for the control-law switching surface. The lunar-module autopilot incorporating the optimal control law received its first flight test in March of 1969. The telemetry from the actual flight indicated that this control law performed very well, in close agreement with the preflight simulation predictions.