Memorandum

Subject: The Advantage of CMGs Over Reaction Wheels To: Dan Kim From: David Bailey Date: October 12, 2016

Summary BMW can keep a bike balanced, if it starts balanced and there are very minor disturbances. Conclusions If BMW isn’t using CMGs, they are likely using reaction wheels to balance their vehicle as referenced by the reaction wheel powered “cubli” seen on their website. Reaction wheels are incredibly energy intensive, while our patented CMG control system is both energy and technologically efficient. There is not enough evidence provided in BMW’s demonstration to conclude that they have developed a balance control with more than very limited authority. Observations The BMW claim that they have a self-balancing motorcycle that does not use Control Moment Gyroscopes (CMGs) is possible. The other angular momentum, transfer device is a reaction wheel. Both were developed for attitude control of satellites. A reaction wheel powered “cubli” on BMW’s website:

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Balancing a two-wheeled vehicle requires angular momentum torque, which is the force that enables the vehicle to lean and self-balance. A reaction wheel applies torque by changing the speed and thereby inertia of a wheel, driven by a motor. The C-1 has a maximum torque available from its CMG pair of 550 foot pounds of torque. To get the same amount of torque using a reaction wheel, the motor changing the speed of the wheel would need to generate 550 foot pounds of torque. The torque from a motor is proportional to motor current. The power in a motor is the current times the voltage. The motor voltage is proportional to the motor speed. If a small reaction wheel is used, the speed changes quickly creating only a moderate torque, and the power demanded by a small wheel also increases quickly. Therefore, for efficiency reasons a large inertia wheel is best to use as a reaction wheel, which is heavy and bulky. A CMG uses a high-speed wheel to produce a large angular momentum. In our case the inertia wheel is rotating at 6000 RPM. The maximum gimbal rate is one and a half radians per second. This means that the gimbal can rotate 180 degrees in just over 2 seconds. The total angular momentum delivered to the C-1 is the same as would be produced by changing the speed of a reaction wheel from + 6000 RPM to -6000RPM in just over 2 seconds. Since the torque delivered to the C-1 is perpendicular to the gimbal axis, the gimbal is easy to turn. A lot of torque is delivered to the vehicle with little power, because the torque is generated by changing the angle of the gyros rather than the inertia created by braking. A satellite using reaction wheels can do a 180 degree maneuver in about 30 minutes, where a satellite using CMGs can do the same maneuver in about 30 seconds. This means that CMGs are used on satellites that are highly agile. A motorcycle needs a lot of torque to balance, and this makes CMGs the correct choice for momentum transfer torque. Lastly, the reference to momentum transfer cubes, as found in BMW’s presentation. An example of a reaction wheel is the MIT cube, a type of momentum transfer cube. The MIT cube, roughly 2 inches on an edge, spins up a small reaction wheel to high speed. The acceleration torque is well below the torque needed to roll the cube over. Once up to speed the wheel is grabbed using a brake, suddenly transferring all the angular momentum from the wheel to the frame. The increased angular momentum of the cube is enough to roll it over to the next side. To do it again the operator has to wait for the wheel to accelerate back up to high speed again. A balancing control can’t wait for a wheel to accelerate to high speed before applying torque to the motorcycle, as would be the case when using reaction wheels. The Cubli balancing cube is similar to the MIT cube, but has an IMU (or inertial measurement unit) to measure orientation and a program that allows it to balance on a corner. This cube uses the braking of the momentum wheel to bring it from its side to and its edge, and then again to its corner. Once on its corner, the reaction wheels stay close to zero speed and are used for balance. BMW used this to show how a reaction wheel is used to balance a cube and thus a motorcycle in a similar way. I don’t think they were implying that they were going to suddenly break their reaction wheel to pop the bike up to a balancing position, although that could be done with a sufficiently large reaction wheel running at a sufficiently high rate of speed.

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If BMW tries to use reaction wheels for control, they will not be very far into development before they change to CMGs because they will not be able to handle the power requirements of a reaction wheel, while getting the performance that would rival Lit Motors C-1. Informational Resources: Control Moment Gyroscopes

● Control Moment Gyroscope on Wikipedia ● Our C-1's CMG dynamics modeled using motorcycle data ● An overview of our CMG tech ● C-1 prototype demo

Reaction Wheels

● Reaction wheel on Wikipedia ● Reaction wheels used in the MIT cube ● Reaction wheels used in the cubli

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