A semi-analytic approach to spacecraft attitude guidance ⋆ Helen C. Henninger a , James D. Biggs a , a Dipartimento di Scienze e Tecnologie Aerospaziale, Politecnico di Milano, Via La Masa 34 - 20156 Milano, Italy Abstract - Attitude slew motions for spacecraft are usually undertaken using feedback control where only the desired fi- nal attitude is stated. In this paper attitude guidance is consid- ered which could be used, in addition to feedback control, to enhance the efficiency of slew motions by pre-planning time- dependent attitude motions. This is achieved using a three- step method in which the angular velocities are expressed as analytic functions in terms of free parameters (on the virtual time domain), and the boundary conditions on the rotation are matched using a shooting method based on a discretized form of Rodrigue’s formula. Following this, the virtual time is reparametrized. This is applied to design a rest-to-rest two- impulse slew manoeuver and a slew motion using only two reaction wheels. Index terms: Attitude control; spacecraft; algebraic and geometric methods 1 I NTRODUCTION This paper is motivated by the prospect of using small space- craft to replace much larger spacecraft to undertake Earth observation and space science missions. In particular, nano- spacecraft (1-10kg) are being considered to significantly re- duce launch costs. However, with any significant reduction in size of a spacecraft there is usually a significant reduc- tion in capability. For example, the attitude control of nano- spacecraft is challenging as they have much harder constraints in terms of fuel usage, either when thrusters are used to con- trol the attitude directly or when they are used to de-saturate reaction wheels in deep-space missions. Furthermore, their ac- tuators are far more likely to fail since the dogma of reliability- based design on large spacecraft is relaxed in favour of effi- ciency design. In the case of attitude slewing, spacecraft use feedback-control where the control is proportional to a prescribed error function between the current attitude of the spacecraft and the desired attitude [1]. Furthermore, the implementation of these con- trollers usually only requires the specification of the final state and not the entire time-dependent motion during the manoeu- vre. Although such proportional-type controls are robust and commonly used on board spacecraft they are not optimal and do not account for actuator faults. In this paper, the use of attitude guidance to pre-plan an efficient motion is considered which can then be tracked to enhance the efficiency of the slew manoeuvre. In this paper, a general semi-analytical method is presented that defines the angular velocities of the spacecraft in the body frame in terms of a number of free parameters that can be selected numerically to match boundary conditions and minimize prescribed cost functions. In our examples nu- merical shooting with different values of these parameters is used to match the boundary conditions on the rotation. Due to the analytical description of the angular velocities, time- parameterization can be used to ensure that the motion is dynamically feasible and potentially further minimize a pre- scribed cost function. The method is used to design an efficient rest-to-rest slew motion for a nano-spacecraft using only two impulses; one delivered by a thruster at the beginning of the motion to initialize it and one at the end to bring it to rest when it arrives naturally at its desired attitude. The method itself is able to calculate the initial and final impulsive torque required to exactly perform the motion efficiently. In addi- tion, the method is used to design a slew motion using only two reaction wheels. In this case the angular velocities are analytically defined in terms of several parameters by solv- ing a non-holonomic optimal kinematic control problem. The boundary conditions are then satisfied by numerical shooting of these parameters. While the applications considered here consider spacecraft at- titude guidance, this method could be applied to several other systems with constraints defined on matrix Lie groups, for ex- ample attitude pointing constraints [2], the wheeled robot with sliding constraints [3], the planar Kirchhoff elastic rod held at each end by a robotic gripper [4] and underwater vehicles [5]. The dynamic equations of motion of a spacecraft controlled ⋆ Corresponding author H. C. Henninger. Tel. +39 3420270338. Email Addresses: helenclare.henninger@polimi.it (Helen C. Henninger), jamesdouglas.biggs@polimi.it (James D. Biggs) 1 2017 25th Mediterranean Conference on Control and Automation (MED) July 3-6, 2017. Valletta, Malta 978-1-5090-4533-4/17/$31.00 ©2017 IEEE 1231