Understanding Gimbal Lock and Its Critical Role in Spacecraft Navigation

Introduction

In aerospace engineering, knowledge of the orientation of a spacecraft, its attitude (A vehicle’s orientation in three-dimensional space, commonly known as attitude) is crucial to a successful operation in space, in terms of navigation, communications and mission. It is a matter of reliable rotating measurements in three independent directions, whether an engineer has to rectify a trajectory in midflight or to point the antenna in the right direction towards the Earth. The effects of the loss of this reliability can be damaging. A phenomenon called a gimbal lock is also one of the most historically important dangers to rotational measurement, a mechanical singularity that has dominated spacecraft design ever since the dawn of human spaceflight.

Understanding Gimbals

A gimbal is a support ring which is pivoted and allows an object mounted on it to move around a single axis freely. Three rings interlocked concentrically make a three-axis gimbal system that can measure or control rotation in the pitch (nose up/down) and yaw (nose left/right) directions and roll (rotation about the longitudinal axis) direction. This is the mechanical heart of an Inertial Measurement Unit (IMU), which ensures that a stable platform within the gimbals can stay in inertial space irrespective of the maneuvers of the spacecraft.

What is Gimbal Lock?

Gimbal lock happens when two of three gimbal axes become coplanar or in other words get rotated in the same geometric plane. At this stage, the system becomes deficient of a degree of rotational freedom: two axes now execute the same tasks, and the rotations around the third one turn to be alike or immeasurable. The system eliminates the three independent axes and forms a mathematical singularity in the equations of orientation.

One way of imagining it is by picturing a book in your hand in a flat position. Typically it can be tilted forward, turned sideways or rotated in place in three free directions. Now turn the book 90 degrees forward so that its spine is now facing you. Then, when one tips it on its side and turns it around, the same happens. A freedom dimension has fallen. This is exactly what occurs in a gimbal lock situation, there are two physical rings that match each other and the system cannot freely solve all the three axes.

Apollo 13 Case

One of the most documented cases of the in-flight navigation difficulties in the history of human spaceflight includes the Apollo 13 mission on April 11, 1970. After the disastrous burst of an oxygen tank about 56 hours into the mission, the crew had to shut down Command Module and use the Lunar Module Aquarius as a lifeboat. This reconfiguration in case of emergency was a big source of navigation difficulties.

The main navigation of the spacecraft was based on Apollo Guidance Computer (AGC) and its IMU. When critical trajectory correction burns were necessary to safely put the crew back on Earth, it was necessary for mission controllers and the crew to monitor and control gimbal angles carefully. The fact that a gimbal lock condition would be approached especially at the time of the critical attitude maneuvers required to complete the PC+2 (a scheduled rocket engine firing performed about two hours after pericynthion: the spacecraft’s closest approach to the Moon) burn presented a risk of corrupting the attitude data as measured by the IMU. The loss of such data would have rendered proper navigation in reentry impossible. Flight controllers and the crew improvised attitude maneuvers that flew dangerously close to gimbal lock limits highlighting the actual risk of operation the phenomenon poses.

Engineering Significance and Modern Solutions

The natural singularity of Euler angle based gimbal systems encouraged the aerospace community to consider formulating alternative mathematical characterizations of orientation. A four component number system called quaternions was proposed in the 19th century by Sir William Rowan Hamilton to give a singularity free definition of rotation, and is now the standard of spacecraft attitude control programs. Quaternions can represent any possible orientation unlike Euler angles, which approach the degenerate case of gimbal lock.

Redundant sensor architectures are also used in modern spacecraft, such as star trackers, sun sensors, and ring laser gyroscopes, which give attitude determination pathways, which are always well defined without reference to gimbal geometry. Attitude representations based on software defined representations have massively replaced the use of mechanical gimbal systems in new designs of missions, although physical gimbals are still being used in antenna pointing and telescope stabilization where gimbal lock avoidance is still an influence on operation processes.

Conclusion

Gimbal lock is one of the basic geometric constraints having far reaching practical implications in spacecraft navigation. Even during the crisis situation, the Apollo 13 mission showed that the consciousness of such a singularity was the key to the survival of the crews. Gimbal lock is hence an obligatory information, not only in the attitude management of spacecrafts, but in any other system that depends upon three axis rotational measurement by robotic arms to virtual reality tracking apparatus, to business aviation.

Written by:

• Fyroze Ripa

• 3rd Year 2nd Semester, Mechanical Engineering (Projjolon - 24th Batch)

• Ahsanullah University of Science and Technology (AUST)

References

• R. R. Bate, D. D. Mueller, and J. E. White, Fundamentals of Astrodynamics. New York, NY, USA: Dover Publications, 1971.

• NASA, "Apollo 13 Mission Report," NASA Technical Report MSC-02680, NASA Manned Spacecraft Center, Houston, TX, USA, 1970. [Online]. Available: https://www.nasa.gov/history/afj/ap13fj/

• D. A. Vallado, Fundamentals of Astrodynamics and Applications, 4th ed. Hawthorne, CA, USA: Microcosm Press, 2013.

• J. R. Wertz, Ed., Spacecraft Attitude Determination and Control. Dordrecht, Netherlands: Kluwer Academic Publishers, 1978.

• W. R. Hamilton, "On Quaternions; or on a New System of Imaginaries in Algebra," Philosophical Magazine, vol. 25, no. 3, pp. 489–495, 1844.

• NASA, "Apollo Guidance, Navigation and Control System," NASA Technical Note TN D-4788, 1968.

• C. Hall, "Spacecraft Attitude Dynamics and Control," Lecture Notes, Aerospace & Ocean Engineering Dept., Virginia Tech, 2003. [Online]. Available: http://www.aoe.vt.edu/~chall/courses/aoe4140/

• J. Diebel, "Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors," Stanford University Technical Report, 2006.

• H. S. F. Cooper, Thirteen: The Apollo Flight That Failed. Baltimore, MD, USA: Johns Hopkins University Press, 1995.

• MIT OpenCourseWare, "16.07 Dynamics," Massachusetts Institute of Technology, 2009. [Online]. Available: https://ocw.mit.edu/courses/16-07-dynamics-fall-2009/

• NASA Jet Propulsion Laboratory, "Attitude and Orbit Control," JPL Technical Publication 1990. [Online]. Available: https://www.jpl.nasa.gov/

• T. Lee, M. Leok, and N. H. McClamroch, "Geometric Tracking Control of a Quadrotor UAV on SE(3)," in Proc. 49th IEEE Conference on Decision and Control, Atlanta, GA, USA, 2010, pp. 5420–5425.