A Review of Active Mechanical Driving Principles of Spherical Robots <s> Scissored-Pairs <s> *† Control-moment gyroscopes (CMGs) are power-efficient attitude-control actuators that produce high torques for agile spacecraft. We propose the use of CMGs in actuating joint degrees of freedom in a spacecraft-mounted agile coelostat telescope, whose tasks include acquiring and tracking a high-speed target. High agility, on the order of several radians per second, is a priority for such a system; however, such capabilities are achieved with traditional actuators only at the expense of excessively high electrical power. The proposed design provides reactionless, agile slewing of a telescope for a small fraction of the power required by fixed rotors in a reaction wheel assembly. This study provides explicit equations of motion for the proposed system and demonstrates by simulation that a CMG-driven system offers the same agility with less than 10% of the power of a telescope actuated by reaction wheels. i/ ω = angular velocity of body i relative to inertial frame 0 i/j ω = angular velocity of body i relative to body frame j i i/j ω = time derivative in reference frame i of i/j ω j i ω = Jacobian matrix of partial angular velocity vectors j i Q = transformation matrix from body j coordinates to body i coordinates i θ = angle of body i relative to body i-1 i φ = gimbal angle of either CMG on body i i e = column matrix of basis vectors for body i ji e ˆ = basis vector in direction i within body j frame i I = inertia dyadic of body i about its center of mass ij G I = gimbal inertia dyadic of CMG j on body i ij R I = rotor inertia dyadic of CMG j on body i C I = inertia dyadic of combined CMG rotor and gimbal C i I = composite inertia dyadic of body i combined with attached CMG scissored pair ij h = rotor angular momentum of CMG j on body i net h = net rotor angular momentum of a general CMG scissored pair i T = sum of the generalized inertia torques in a system a i T = sum of the generalized active torques in a system a i τ = torque applied to the system along direction i <s> BIB001 </s> A Review of Active Mechanical Driving Principles of Spherical Robots <s> Scissored-Pairs <s> Control-moment gyroscopes (CMGs) are power-e‐cient attitude-control actuators that produce high torques on spacecraft. In this study, a method is investigated for maneuvering robotic arm segments in a spacecraft-mounted, CMG-actuated agile imaging payload with minimal power usage. A real-time optimization method is presented that includes null motion in a closed-loop end-efiector tracking problem. The proposed approach involves flrst establishing kinematic relationships between the end-efiector attitude coordinates and joint coordinates. These transformations are used to track a two-coordinate end-efiector attitude command by simultaneously controlling three joint degrees of freedom. With the redundant degree of freedom, there are multiple joint-angle solutions corresponding to a given boresight attitude command. While tracking this command, null motion is added to the commanded joint angles. The augmented joint-angle command causes a change in the body and gimbal motion that ultimately reduces power consumption. The null-angle component is determined by computing the null projection of the cost gradient with respect to the joint angles. A quadratic cost function is deflned, which is the sum of the squares of power for each CMG gimbal. Simulation results demonstrate that the power consumption of the system, when measured by the integral of non-recoverable power, is reduced by up to 42% when a certain amount of null motion is included in the feedback loop. <s> BIB002
The idea of using gyroscopes for conservation of angular momentum based spatial control has been utilized to control large space structures such as the International Space Station and to stabilize large telescopes BIB001 . More complex designs can even be used as inertial dampers in space. Based on the mission, four to six variable-speed control moment gyroscopes can be configured in such a system  . As mentioned before, use of control moment gyroscopes has an advantage and a disadvantage: an action on the gyroscope causes a reaction in two orthogonal planes of space. This means that unless this is accounted for in the design, it can cause unwanted problems. One novel way to circumvent this issue is the use of a scissored-pair ( Figure 21 ). By pairing each control moment gyroscope with another that is spinning in the opposite direction at the same magnitude (h 1 and h 2 ), the reaction from the changing spin of the gyroscopes cancel each other out and now an action on gyroscope pair (τ 1 and τ 2 ), causes a reaction in only one direction (h net ). However, one downside of this configuration is that h net switches direction after θ becomes 180 • or more. After that, the direction of h net is pointed in the opposite direction, and goes back to the original direction when θ becomes zero. Spinning the CMGs down to zero angular velocity or spinning the CMGs in the opposite direction can neutralize this phenomenon. This configuration has recently been used in space robotics applications for low power actuators  , and even more recently been used in augmenting spherical robot control. Usage of such a device is exhibited in G. Schroll's Master's thesis as well as patent   . The design won him the Collegiate Inventors Competition in 2008, and he was the first undergrad to ever receive the award [38] . The robot can be thought of as an enhanced pendulum-type design. It incorporates a variable-speed scissored pair into its bob, allowing it to have a torque greater than mgr * sin(θ) (shown in Figure 1 ). The output torque can be thought of as where h net is the torque from the scissored pair, m is the mass of the bob, and r is the radius of the bob from the center of the robot. Since h net is a function of the angular velocity of the CMG pair, the output torque is no longer dependent on the weight of the bob, proverbially allowing the center of mass to be shifted outside of the shell. Videos show the robot climbing out of steep holes and making sharp turns. Although the control system is still being researched, this is a viable solution to the torque limit analogous with pendulum-based designs. Figure 21 . Picture of a Scissored-Pair configuration BIB002 . Figure 22 . G. Schroll's spherical mobile robot internal assembly and prototype   . Although this is an innovative approach, it is only a momentary boost of torque and does not provide a continuous driving force. As mentioned above, the torque boost only lasts for a rotation of 180 • . After that boost, the CMGs must be reconfigured either through mechanical or control methods in order to prevent an unwanted secondary boost in the opposite direction. This presents a whole new area of research for spherical robots, as there are many possibly ways to achieve such an effect: gearing, controlling spin of the gyroscope speed, etc. A proper control method or mechanical design to change the output torque direction of a scissored pair from an oscillating to a uni-directional state may lead to a highly advanced spherical robot design.