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Aerospace
, Volume 6 (9) – Sep 4, 2019

/lp/multidisciplinary-digital-publishing-institute/non-symmetric-gyroscope-skewed-pyramids-AVQNLvTI2T

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- Multidisciplinary Digital Publishing Institute
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- 2226-4310
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- 10.3390/aerospace6090098
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aerospace Article 1 ,y 1 ,y 1 ,y 1 ,y 1 ,y Zachary Lewis , Joshua Ten Eyck , Kyle Baker , Eryn Culton , Jonathan Lang and 2 , ,y Timothy Sands * Department of Mechanical and Aerospace Engineering, Naval Postgraduate School, Monterey, CA 93943, USA Department of Mechanical Engineering (CVN), Columbia University, New York, NY 10027, USA * Correspondence: dr.timsands@caa.columbia.edu y These authors contributed equally to this work. Received: 8 July 2019; Accepted: 29 August 2019; Published: 4 September 2019 Abstract: The novel contribution in this manuscript is an expansion of the current state-of-the-art in the geometric installation of control moment gyroscopes beyond the benchmark symmetric skewed arrays and the four asymmetric arrays presented in recent literature. The benchmark pyramid symmetrically skewed at 54.73 degrees mandates signiﬁcant attention to singularity avoidance, escape, and penetration, while the most recent four asymmetric arrays are strictly useful in instances where space is available to mount at least one gyro orthogonal to the others. Skewed arrays of gyros and the research-benchmark are introduced, followed by the present-day box-90 and “roof” conﬁgurations, where the roof conﬁguration is the ﬁrst prevalently used asymmetric geometry. Six other asymmetric options in the most recent literature are introduced, where four of the six options are obviously quite useful. From this inspiration, several dozen discrete options for asymmetric installations are critically evaluated using two ﬁgures of merit: maximum momentum (saturation) and maximum singularity-free momentum. Furthermore, continuous surface plots are presented to provide readers with countless (inﬁnite) options for geometric installations. The manuscript ﬁrmly establishes many useful options for engineers who learn that the physical space on their spacecraft is insucient to permit standard installations. Keywords: U.S. Patent 9567112 B1; control moment gyroscope; actuators; attitude control; guidance; navigation and control; spacecraft dynamics; skewed pyramid 1. Introduction Motion mechanics is governed by Euler ’s moment equations for three degrees of rotation of rigid bodies in space [1], and Newton’s Law [2] for three dimensions of translation. External forces produce a change of linear momentum (i.e., a change in velocity with time for a constant mass), while external torques produce a change in angular momentum (i.e., a change in angular velocity with time for a constant mass moment of inertia). Rigid body motion mechanics governing spacecraft attitude control has a long, distinguished lineage of literature [3–17] that has culminated in recent advances using the governing Euler ’s moment equations as the control [18–27] in a scheme called deterministic artiﬁcial intelligence [28,29]. One alternative to generating an external torque is to modify the angular momentum of some part(s) of the spacecraft, imparting motion for the entire system whilst maintaining conservation of the angular momentum. Control moment gyroscopes [30] are an example of such a system. Gyros use a relationship called a steering law [31–36] to calculate gyro commands for designed changes in angular momentum (torque commands generated by attitude controllers to accomplish a desired maneuver). Engineers design maneuvers in such a manner so that they never demand torque that would produce an angular momentum trajectory that is required to pass through Aerospace 2019, 6, 98; doi:10.3390/aerospace6090098 www.mdpi.com/journal/aerospace Aerospace 2019, 6, 98 2 of 14 a singular condition [37–47]. One of the methods used by engineers is to analyze and implement gyro installation at angles that yield the most usable momentum space [48], where usability is deﬁned as singularity-free operations yielding the maximum angular momentum. This manuscript introduces the control moment gyroscope as a spacecraft torque actuator in the benchmark installation geometry (symmetric arrays of gyros) modiﬁed to a non-redundant instantiation. The initial research on non-symmetric skewed arrays is brieﬂy reviewed to illustrate the usefulness of the concept, since the introduction of asymmetry rotates the momentum space, allowing increased singularity-free torque production in a desired direction. Ubiquitous skew angle optimization for singularity-free operations results in symmetric arrays of gyros skewed at ninety degrees. From this baseline, depressing individual gyros from ninety to zero degree skew angles has already demonstrated n 2 momentum space rotation in the literature [43]. These 3 = 3 = 9 instantiations are the state-of-the-art for asymmetric skewed arrays. From this inspiration, this manuscript investigates many of the inﬁnite possibilities for skew angle combinations. The ﬁgures of merit for comparison are the maximum singularity-free momentum and maximum (saturation) momentum capabilities. The originally sought research objective was to increase the yaw maneuvering capability, and several iterations towards this goal are presented. In light of recent research illustrating the peculiarities of near-planar momentum associated with low skew angles [48], where two gyros are held in-plane at a zero (and near-zero) degree skew angle, while the third gyro is iterated. Subsequently, only one gyro is held at a very low (non-zero) skew angle, while iterating the other two. Pseudo-cardinal quantities (from a sinusoidal perspective) are also investigated by iterating each gyro through a zero, thirty, sixty, and ninety degrees skew angle. Other instantiations investigated include ﬁfteen, twenty, thirty, forty, forty-ﬁve, sixty, eighty, and ninety. Taken together, the manuscript comprises a novel initial articulation of asymmetric skewed arrays permitting guidance, navigation, and control engineers to participate in an integrated product team where other constituent members may enforce physical space constraints that preclude using the new-benchmark “box-90” conﬁgurations of symmetric gyros skewed at ninety degrees. 2. Materials and Methods This study will be limited to non-redundant, constant-speed, single-gimbaled control moment gyroscopes (CMGs). Well-established methods are described ﬁrst to elaborate typical skewed arrays of gyros (ubiquitous “benchmark” geometry [42]), inverse steering laws, and the gyro system matrix (referred to as [A] in the literature). The matrix contains time-varying gimbal angles that may be commanded to produce a desired torque, and the matrix also contains skew angles representing the gyro installation angle relative to the spacecraft reference frame. Changing these skew angles can bring about unique attributes for the spacecraft, such as an increased pitch, roll, or yaw capability. Mapping out these singularities for iterated gyro mounting geometries can show an engineer how to maximize these capabilities and enhance a spacecraft’s mission completion ability. 2.1. Benchmark Redundant Arrays and Non-Redundant Arrays Figure 1b depicts a rigid spacecraft with four gyros (Figure 1a) described in [43], in the so-called benchmark conﬁguration [42], where redundancy is from four gyros for three degrees of freedom embodied in three-dimensional motion. The spacecraft’s body reference frame is labeled x, y, and z, and each gyro is mounted at a ﬁxed skew angle of 8i = 1, 2, 3, 4 = during installation. 2.2. State-of-the-Art Non-Redundant Symmetric Skewed Arrays Typical non-redundant symmetric installation is depicted in Figure 2a, while recently developed asymmetric installation is depicted in Figure 2b. Skew angles are not normally modiﬁed after installation. The gyros are gimballed, and their gimbal rotations about their respective gimbal axes are expressed as 8i = 1, 2, 3. Gimballing rotates the angular momentum vectors h 8i = 1, 2, 3 in i i planes perpendicular to the gimbal axis, and these planes contain the respective torque vectors in a Aerospace 2019, 6, 98 3 of 14 Aerospace 2019, 6, x FOR PEER REVIEW 3 of 15 right-hand arrangement: angular momentum, torque, and the gimbal axis. The torque axis () rotation for a CMG is the axis around which the spacecraft maneuver is accomplished. The torque axis and the rotation for a CMG is the axis around which the spacecraft maneuver is accomplished. The torque orthogonal relationship between the CMG’s gimbal axis () and angular momentum axis (h ) can be axis and the orthogonal relationship between the CMG’s gimbal axis (𝛿 ) and angular momenti um axis seen in Figure 1. These relationships demonstrate how one may discover the torque axis and provide (ℎ ) can be seen in Figure 1. These relationships demonstrate how one may discover the torque axis the intuition behind singularity generation in this conﬁguration [48]. and provide the intuition behind singularity generation in this configuration [48]. (b) Ubiquitous symmetrically skewed array of four gyros (a) Single-gimballed control moment gyroscope (minimally redundant) (c) CMG mounted at a skew angle of (d) CMGs mounted in a vertical space-saving configuration 𝛽 57 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 Figure 1. Control moment gyroscopes (a) placed onto a spacecraft mounted at a skew angle, with 𝛽 Figure 1. Control moment gyroscopes (a) placed onto a spacecraft mounted at a skew angle, with depicted here (b) in the former research benchmark with minimally redundant (four) gyros arranged depicted here (b) in the former research benchmark with minimally redundant (four) gyros arranged in in the shape of a pyramid skewed at the symmetric angle, 𝛽 . Notice the reduced vertical space the shape of a pyramid skewed at the symmetric angle, . Notice the reduced vertical space occupied occupied by the gyros in (d) compared to the vertical space occupied by the gyro in (c). This vertical by the gyros in (d) compared to the vertical space occupied by the gyro in (c). This vertical space space reduction is one reason why engineers might need to understand configurations with one or reduction is one reason why engineers might need to understand conﬁgurations with one or more gyro more gyro mounted at disparate skew angles resulting in an asymmetric configuration. mounted at disparate skew angles resulting in an asymmetric conﬁguration. Figur Figure e 2 2 depicts depicts a a rig rigid id sp spacecraft acecraft with with only only thr three gy ee gyr ros (non-redundancy os (non-redundancy for for th thr ree degre ee degrees es o off freedom embodied in three-dimensional motion) [44]. Frames and axes are consistently labeled with freedom embodied in three-dimensional motion) [44]. Frames and axes are consistently labeled with rregar egards ds to th to the e minimally minimall ry redund edundantan case t ca inse Figur in Fig e 1,udespite re 1, despit lacking e la acki fourth ng agyr foourt (the h gyro (t non-redundant he non- redundant case). Notice the literature’s [43,44] use of a capitalized 𝐻 to denote an individual gyro’s case). Notice the literature’s [43,44] use of a capitalized H to denote an individual gyro’s angular momentum angular mome (seen ntu in mFigur (seen e i1 n ), F whil igure e we 1), wh prefer ile w the e p use refof er t ah lower e use o -case f a lterm owerassembled -case term a in ss the emcapitalized bled in the n o capitalized vector (seen in Figure 2) 𝐻= ℎ ℎ ℎ . Notice that vectors annotated in bold font in vector (seen in Figure 2) H = h h h . Notice that vectors annotated in bold font in the ﬁgures 1 2 3 ar the e also figure written s are ain lso w equations ritten in e using quation { } brackets, s using { underlined } brackets, under terms,lin ored terms, or e even over-arr v ows. en over-arrows. Aerospace 2019, 6, 98 4 of 14 Aerospace 2019, 6, x FOR PEER REVIEW 4 of 15 (a) Non-redundant array skewed (b) Depiction of a permissible plane of rotation symmetrically at 𝛽 = 𝛽𝛽 𝛽 of an angular momentum vector Figure 2. Spacecraft body axes serving as basis vectors in schematic drawings of skewed SGCMG Figure 2. Spacecraft body axes serving as basis vectors in schematic drawings of skewed SGCMG array array geometries, noting the angular momentum, torque, and gimbal vectors that define a right-hand geometries, noting the angular momentum, torque, and gimbal vectors that deﬁne a right-hand set. set. . . According to Euler ’s equation T = H = H + ! H , a time-derivative (rate of inertial inertial B B B According to Euler’s equation ∑ 𝑇 =𝐻 =𝐻 𝜔 𝐻 , a time-derivative (rate of change) of the angular momentum is tor que, her e referred to as the (negative) torque of the CMGs, change) of the angular momentum is torque, here referred to as the (negative) torque of the CMGs, where is referred to in typical literature on controls as the variable u. Equation (1) is the steering CMG where 𝜏 is referred to in typical literature on controls as the variable 𝑢 . Equation (1) is the law (in the forward direction) that is merely the time-derivative of the projection of the individual steering law (in the forward direction) that is merely the time-derivative of the projection of the angular momentum vector H = fh h h g . The relationship is expanded using the product rule 1 2 3 individual angular momentum vector 𝐻= ℎ ℎ ℎ . The relationship is expanded using the for derivatives to express the change in angular m omentum as the product of the gimbal rates and a product rule for derivatives to express the change in angular momentum as the product of the gimbal spatial gradient Jacobian matrix containing gimbal angles and skew angles referred to in the literature rates and a spatial gradient Jacobian matrix containing gimbal angles and skew angles referred to in as [A]. (ASIDE: notice the change in nomenclature from Figure 1, the benchmark conﬁguration, where the literature as [𝐴 ]. (ASIDE: notice the change in nomenclature from Figure 1, the benchmark a lower-case h is assembled into an upper-case vector H). The relationship must be inverted, since the configuration, where a lower-case hi is assembled into an upper-case vector H). The relationship must control u is known, and the current known gimbal positions deﬁne the [A] matrix, while the necessary be inverted, since the control 𝑢 is known, and the current known gimbal positions define the [𝐴 ] gimbal commands are unknown. Equation (2) reveals the inverse relationship for the gimbal steering matrix, while the necessary gimbal commands 𝛿 are unknown. Equation (2) reveals the inverse commands, where control u is generated by typical equations of automatic control (e.g., a feedforward relationship for the gimbal steering commands, where control 𝑢 is generated by typical equations of controller like nonlinear optimal control, and feedback controllers like PI-Proportional, Integral, automatic control (e.g., a feedforward controller like nonlinear optimal control, and feedback PD-Proportional, Derivative, and PID-Proportional, Integral, Derivative controllers, etc.). The matrix controllers like PI-Proportional, Integral, PD-Proportional, Derivative, and PID-Proportional, inverse in Equation (2) is usually a matrix pseudoinverse in the literature, since the benchmark research Integral, Derivative controllers, etc.). The matrix inverse in Equation (2) is usually a matrix conﬁguration has four gyros providing four degrees of freedom for only three dimensions of motion pseudoinverse in the literature, since the benchmark research configuration has four gyros providing (i.e., an overdetermined problem for which the pseudo-inverse is appropriate). The regular matrix four degrees of freedom for only three dimensions of motion (i.e., an overdetermined problem for inverse is correctly used here for this precisely determined problem where three degrees of freedom which the pseudo-inverse is appropriate). The regular matrix inverse is correctly used here for this are used to provide three-dimensional motion. The [A] matrix is the spatial gradient matrix of angular precisely determined problem where three degrees of freedom are used to provide three-dimensional momentum normalized by one gyro’s value of angular momentum, as per Equations (3) and (4). motion. The [A] matrix is the spatial gradient matrix of angular momentum normalized by one gyro’s . . dH dH d value of angular momentum, as per Equations (3) and (4). u = H = = [A] (1) CMG dt d dt 𝑑𝐻 𝑑𝐻 𝑑𝛿 = [A] u [ ] (1 (2) ) 𝜏 ≡𝑢 =𝐻 = = ≡ 𝐴 𝛿 𝑑𝑡 𝑑𝛿 𝑑𝑡 8 9 8 9 > > > > h cos cos + cos sin x 3 1 2 > > > > > > > > < = < = 𝛿 = [𝐴 ] 𝑢 (2) H = h = cos (sin sin ) cos (3) > > > > y 3 1 2 > > > > > > > > : ; : ; h cos (sin + sin + sin ) z 2 3 ℎ 𝑐𝑜𝑠𝛿 𝑐𝑜𝑠𝛿 𝑐𝑖𝑛𝑠𝛽𝑜𝑠𝛿 2 3 ℎ ( ) 𝐻= =6 cos sin𝛽 𝛿 cos 𝑠𝑖𝑛𝛿 cos 𝑐𝑜𝑠𝛿 sin 7 (3) 1 2 3 6 7 6 7 dH 6 7 6 7 ℎ cos 𝛽 (𝛿 𝑠𝑖𝑛𝛿 𝑠𝑖𝑛𝛿 ) [A] = = cos cos sin cos cos (4) 6 7 1 2 3 6 7 6 7 4 5 sin cos sin cos sin cos 1 2 3 𝛿 𝑐𝑜𝑠𝛽𝑐𝑜𝑠𝛿 𝑠𝑖𝑛𝛿 𝑑𝐻 It should be noted that the skew angles 8i = 1, 2, 3 = are also in the [A] matrix in Equation (4) [ ] 𝑠𝑐𝑜𝑠𝛽𝑐𝑜𝛿 i 𝛿 𝑐𝑜𝑠𝛽𝑐𝑜𝑠𝛿 𝐴 = = (4) 𝑑𝛿 due to its use of projecting the angular momentum vectors of the gyros onto the spacecraft reference 𝑠𝑖𝑛𝛽𝑐𝑜𝑠𝛿 𝑠𝑖𝑛𝛽𝑐𝑜𝑠𝛿 𝑠𝑖𝑛𝛽𝑐𝑜𝑠𝛿 𝑠𝑖𝑛 𝑠𝑖𝑛 𝑠𝑖𝑛 𝑠𝑖𝑛 Aerospace 2019, 6, 98 5 of 14 frame, as done in Equation (3). Therefore, contributes to mathematical singularity, since the [A] must be inverted in Equation (2). The inverse itself is of interest in this study, and several methods of computing the inverse were investigated in this research for a non-redundant (three gyro) array, and the results are summarized in Table 1. While an analytic inverse is rarely used for overdetermined systems in favor of the pseudoinverse, since the steering matrix has a [3 3] dimension in this non-redundant case, analytical expressions are available in both matrix form and as decoupled, multiple equations. These are respectively named “analytic inverse” and “analytic formulas” in Table 1. Table 1. Matrix 2-norm error deﬁned as the dierence between each case and the baseline Moore–Penrose matrix pseudoinversion, pinv. Skew angle was set to = f90 90 90g during a 30 degree maneuver, while gimbal angles = f g were free to rotate. i 1 2 3 [A] Inversion Method Matrix-Norm Error 0.872826646563208 [A] [ ] inv A 0.872826646563208 pinv[A] – [A]neye(size([A])) 0.872826646560334 LU Decomposition 0.872826646560693 Analytic Matrix 6.67204727298 10 Analytic Formulas 6.67204727298 10 Notice identical results for [A] and inv[A] implementations. A common controversial issue is that the chosen computational implementations produce disparate results, despite using the same equation. One such computational implementation is the methodology chosen to calculate the matrix inverse (the source of mathematical singularities) in the steering law in Equation (2). In order to aid reproducibility of the results in this manuscript, seven common inversion methods are compared in Table 1, while the Moore–Penrose pseudoinverse was chosen here and implemented by the MATLAB pinv command. The Moore–Penrose pseudoinverse implemented in MATLAB was the baseline for comparison, since it had the lowest matrix norm error. 2.3. Benchmark Geometry: Optimal Symmetric Skew Angle The minimally redundant (4-gyro skewed pyramid) has been extensively studied in the literature, with the consensus that the skew angle 8i = 1, 2, 3, 4 = = 54.73 degrees [42] establishes a spherical momentum envelope (maximal obtainable angular momentum). This skew angle is the benchmark from which subsequent research has grown in the past few decades, including the re-optimization of the skew angle [45] for the non-redundant to be introduced in the next section of this manuscript. The basis for the original benchmark optimization was to optimize the spherical nature of the outer momentum generation capability (i.e., saturation). Additional knowledge on this aspect of the benchmark geometry is revealed later in this manuscript, when the asymmetric geometries are evaluated for the ﬁrst time for a maximal (saturation) performance. 2.4. State-of-the-Art Geometry: Re-Optimized Symmetric Skew Angle The current state-of-the-art for gyro-pyramid installation geometry comes from a re-optimization of spherical singularity-free angular momentum, and is often referred to in industry as the slang term “box-90”. Notice the substantial dierence between the displays in Figures 3 and 4 of the pyramid singularity hypersurfaces. This heuristically illustrates the beneﬁt of the bi-planar generation of angular momentum, while [48] reveals some weaknesses with the counter-opposing philosophy, near co-planar angular momentum generation. Another conﬁguration that has recently arisen to be commonplace in industry is a “roof conﬁguration”, which will now be elaborated. Aerospace 2019, 6, x FOR PEER REVIEW 6 of 15 Aerospace 2019, 6, 98 6 of 14 Aerospace 2019, 6, x FOR PEER REVIEW 6 of 15 Figure 3. Maximum Singularity Free Momentum of β1 = 90°, and β2, and β3 = Free. Figure 3. Maximum Singularity Free Momentum of = 90 , and , and = Free. 1 2 3 Figure 3. Maximum Singularity Free Momentum of β1 = 90°, and β2, and β3 = Free. (b) Numerical Maximum Singularity Free (a) Numerical Maximum Singularity Free (b) Numerical Maximum Singularity Free Momentum of β1 = 30°, and β2 and β3 = Free (a) Numerical Maximum Singularity Free Momentum of β1 = 0°, and β2 and β3 = Free Momentum of β1 = 30°, and β2 and β3 = Free Momentum of β1 = 0°, and β2 and β3 = Free (d) Numerical Maximum Singularity Free (c) Numerical Maximum Singularity Free (d) Numerical Maximum Singularity Free (c) Numerical Maximum Singularity Free Momentum of β1 = 90°, and β2 and β3 = Free Momentum of β1 = 60°, and β2 and β3 = Free Momentum of β1 = 90°, and β2 and β3 = Free Momentum of β1 = 60°, and β2 and β3 = Free Figure 4. Numerical Maximum Singularity Free Momentum of β1 = 0°, 30°, 60°, and 90°, and β2 and β3 Figure 4. Numerical Maximum Singularity Free Momentum of = 0 , 30 , 60 , and 90 , and and 1 2 Figure 4. Numerical Maximum Singularity Free Momentum of β1 = 0°, 30°, 60°, and 90°, and β2 and β3 = Free. = Free. = Free. 2.5. State-of-the-Art Asymmetric Geometry: Iterated Non-Symmetric Orthogonal Skew Angles 2.6. State-of-the-Art Asymmetric Geometry: Iterated Non-Symmetric Orthogonal Skew Angles 2.6. State-of-the-Art Asymmetric Geometry: Iterated Non-Symmetric Orthogonal Skew Angles The depiction of angular momentum planes established by the use of “mixed” skew angles results The depiction of angular momentum planes established by the use of “mixed” skew angles The depiction of angular momentum planes established by the use of “mixed” skew angles in asymmetries, while the slang “roof conﬁguration” is obviously applicable when the plane for gyro results in asymmetries, while the slang “roof configuration” is obviously applicable when the plane results in asymmetries, while the slang “roof configuration” is obviously applicable when the plane #2 rotates from 90 degrees to 0 degrees (establishing a “roof/ﬂoor” on the seemingly walled-structure). for gyro #2 rotates from 90 degrees to 0 degrees (establishing a “roof/floor” on the seemingly walled- for gyro #2 rotates from 90 degrees to 0 degrees (establishing a “roof/floor” on the seemingly walled- While the box-90 conﬁguration has a maximal capability around the y-axis (pitch), the investigation structure). While the box-90 configuration has a maximal capability around the y-axis (pitch), the structure). While the box-90 configuration has a maximal capability around the y-axis (pitch), the described in [43] consecutively analyzed installations where each gyro(s) is/are rotated from 90 degrees investigation described in [43] consecutively analyzed installations where each gyro(s) is/are rotated investigation described in [43] consecutively analyzed installations where each gyro(s) is/are rotated to 0 degrees. The result is rotation of the singularity hypersurfaces, where the “roof” conﬁguration from 90 degrees to 0 degrees. The result is rotation of the singularity hypersurfaces, where the “roof” from 90 degrees to 0 degrees. The result is rotation of the singularity hypersurfaces, where the “roof” depresses gyro #2 from 90 to 0 degrees, and the maximal capability is established in the z-direction (yaw). configuration depresses gyro #2 from 90 to 0 degrees, and the maximal capability is established in the configuration depresses gyro #2 from 90 to 0 degrees, and the maximal capability is established in the z-direction (yaw). z-direction (yaw). Aerospace 2019, 6, 98 7 of 14 Six instances of non-symmetric skewed pyramids are the current state-of-the-art, but only four options have immediately obvious beneﬁts. The other four subplots are desirable installations for engineers who learn that their space missions need a predominant capability around the roll, pitch, or yaw, respectively, as listed in Table 2. This legacy research inspires new developments in this manuscript. What should an engineer do if they learn that their space mission requires predominant capabilities about some intermediate axis (e.g., an axis in the x-z plane)? What if (for example) a gyro’s mounting space gets prohibitively restricted? Table 2. Mixed skew angles in the literature and resulting preference dictions [43]. CMG Skew Angle Singularity Case Preferred Direction CMG #1 CMG #2 CMG #3 Case 1 0 90 90 NA Case 2 90 0 90 z Case 3 90 90 0 NA Case 4 0 0 90 x ˆ Case 5 0 90 0 y ˆ Case 6 90 0 0 x ˆ CMG skew angles in degrees. 2.6. Methodology There does not exist a reference volume that includes many, a dozen, or continuous inﬁnite selections of installation geometries (options for skew angles, ), and Section 3 accomplishes this key novel contribution to the ﬁeld. Towards the goal of creating such a reference volume, Section 3 of this manuscript reveals the results of our research starting from the current state-of-the-art summarized in Table 2 that has already been presented in the literature. We commence with holding gyro #1 at a 90-degree skew angle and allowing gyros #2 and #3 to vary continuously from 0 to 90 degrees. Similarly, we examine holding gyro #1 at zero degrees and permitting the other two gyros to be installed at any arbitrary angle. Next, we step through logical iterations: what happens when two gyros are ﬁxed and a third is free. Considering recent companion research [48] by the authors illustrating peculiarities of near-planar momentum generation associated with very low skew angles, and following the logical iteration methodology, we next consider holding two gyros at very low skew angles. The aforementioned iterations reveal the maximum singularity-free momentum space available (an inner momentum space), so lastly, we investigate the maximum (saturation) capability, regardless of inner singularities akin to the original benchmark optimization [42]. 3. Results We have introduced the benchmark geometry (Figures 1 and 3) and its corollary state-of-the art non-redundant modiﬁcation (Figures 2, 4 and 5), in addition to initial implementations of asymmetric arrays. The next section’s motivation is the fact that engineers may not always be permitted to install one or more gyro skewed at ninety degrees, occupying the most physical space of all options. Consider a gyro of an arbitrary dimension of unity. Symmetric mounting of three or four such gyros occupies one unit in the vertical plane and also one unit in the horizontal plane. Meanwhile, depressing either of the planes to non-orthogonal mounting at less than ninety degrees requires space of less than one unit. The spinning mass remains in the same dimension, as does the gimbal and mounting hardware, but we rotate that dimension away from an orthogonal plane, reducing the space occupied by the apparatus. Therefore, symmetric, orthogonal mounting at ninety degrees requires the most space onboard a spacecraft compared to any of the other options. Quickly peak back to Figure 1b and notice how depressing the skew angle reduces the height of the pyramid. Similarly, the long-axis of the CMG (depicted in Figure 1a) depresses, reducing the physical size requirement inside the spacecraft. Aerospace 2019, 6, 98 8 of 14 Space systems engineers might be required to install one of more gyros at an angle of less than ninety degrees to (for example) make physical space available to another subsystem (e.g., a cold gas thruster tank, propellant piping, heat exchangers, etc.). Novel, Non-Symmetric, Non-Orthogonal Skew Angles With the understanding that all references to angular momentum are normalized by one gyro’s angular momentum capability (i.e., so-called 1h, 2h, and 3h nomenclature ubiquitous in the literature), we commence the investigation by holding gyro #1 at a 90-degree skew angle and allowing gyros #2 and #3 to vary continuously from 0 to 90 degrees (depicted in Figure 3). The results are self-validating by examining the singularity-free maximum momentum capability for the symmetric case (since it is one of the inﬁnite iterations). 8i = 1, 2, 3 = = 90 degrees is illustrated in Figure 3 to produce 1h, which is one gyros worth of angular momentum, and this exactly coincides with the contemporary literature on the “box-90” conﬁguration. Due to the diculties in visualizing all the possible iterations of angles in Figure 3, we “ﬂatten” the depictions to two-dimensions, necessitating multiple plots depicted in Figure 4 for four semi-cardinal values for the skew angle of gyro #1. Inspired by the eectiveness of momentum space rotation of the “roof” conﬁguration established by laying down the center gyro (gyro #2 here) to zero degrees, companion research by the authors [48] investigated issues that arise when using other combinations of low skew angles. Figure 5 is inspired by the companion research which claims peculiarities with low-skew angle combinations (albeit the “roof” conﬁguration has proven quite eective). We take gyro #1 as the illustrative example, since it has a counterpart, gyro #3, which acts in a parallel plane. Immediately evident in Figure 5a is the fact that laying gyro #1 to a near-zero skew angle does not bestow a beneﬁt unless one of the other gyros is also depressed (allowing near 1h capabilities). Notice how poorly the conﬁguration performs if only gyro #1 is depressed, leaving the other two gyros at ninety degrees. As a matter of fact, the only favorable regions of Figure 5a are in the upper-left and lower-right corners, where one of the other respective gyros is also depressed to zero degrees. Another aspect evident in Figure 5 is the nature of the right hand side of each subplot. Notice that the far-right sides of the plots become only yellow (best performing) when skew angles all are beyond 60 degrees (precisely validating the discovery in the companion research that revealed peculiarities of near-planar momentum associate with low skew angles). Figure 6 presents a three-dimensional version of Figure 5 (gyro #1 held at 1 degree) akin to Figure 4’s original presentation of gyro #1 held at ninety degrees. Meanwhile, Figure 6 moves away from singularity-free capability to focus on maximal (saturation) capability, but the iterations are similar: we iterate the skew angle of gyro #1 discretely, and present continuous (inﬁnite) iterations of the other two skew angles. Table 3 contains the maximum yaw maneuverability skew angle combinations. The space mission motivating this research needed to maximize yaw maneuverability. While the analysis displayed in Figures 4–6 provides (inﬁnite) continuous optional alternatives for mounting gyros asymmetrically, the graphic depictions nonetheless can make it dicult to discern the exact capability of a discrete combination of asymmetric skew angles. To remedy this deﬁciency, the analyses of discrete options are listed in Tables 4–7. For example, if a space mission dictates two that gyros must be set to a skew angle lower than ninety degrees, several discrete options are immediately evident, and some even maintain a very similar singularity-free momentum capability (e.g., = f80 1 1g yields 097h, while the box-90 conﬁguration = f90 90 90g yields 1h). Aerospace 2019, 6, 98 9 of 14 Aerospace 2019, 6, x FOR PEER REVIEW 9 of 15 (a) Numerical maximum singularity-free (b) Numerical maximum singularity-free momentum of β1 = 1°, where β2 and β3 are free momentum of β1 = 10°, where β2 and β3 are free (c) Numerical maximum singularity-free (d) Numerical maximum singularity-free momentum of β1 = 20°, where β2 and β3 are free momentum of β1 = 30°, where β2 and β3 are free (f) Numerical maximum singularity-free (e) Numerical maximum singularity-free momentum of β1 = 50°, where β2 and β3 are free momentum of β1 = 40°, where β2 and β3 are free (g) Numerical maximum singularity-free (h) Numerical maximum singularity-free momentum of β1 = 60°, where β2 and β3 are free momentum of β1 = 70°, where β2 and β3 are free (i) Numerical maximum singularity-free (j) Numerical maximum singularity-free momentum of β1 = 80°, where β2 and β3 are free momentum of β1 = 90°, where β2 and β3 are free Figure 5. Numerically-calculated maximum singularity-free momentum. Figure 5. Numerically-calculated maximum singularity-free momentum. Aerospace 2019, 6, 98 10 of 14 Aerospace 2019, 6, x FOR PEER REVIEW 10 of 15 Figure 6. 3-D maximum (saturation) angular momentum on the z axis for = 1 , where and Figure 6. 3-D maximum (saturation) angular momentum on the z axis for β1 = 1 1°, where β2 and 2 β3 = = free. free. 3 Table 3. Maximum (saturation) yaw maneuverability skew angle combinations. Table 3. Maximum (saturation) yaw maneuverability skew angle combinations. |H| 1 2 3 |H| β1 β2 β3 2.017 1 90 90 2.017 1 90 90 2.259 15 90 90 2.259 15 90 90 2.5 30 90 90 2.5 30 90 90 2.707 45 90 90 2.866 2.70760 45 90 90 90 90 2.966 75 90 90 2.866 60 90 90 3 90 90 90 2.966 75 90 90 Skew angles measured in degrees. 3 90 90 90 Skew angles measured in degrees. Next, we iterated only gyro #1’s installed skew angle, retaining other skew angles at zero degrees. Next, we iterated only gyro #1’s installed skew angle, retaining other skew angles at zero We investigated one-thousand iterations of mixed skew angles, and some key iterations are described degrees. We investigated one-thousand iterations of mixed skew angles, and some key iterations are here. Once again, the maximum amplitude is 1h, representing one gyro’s worth of angular momentum, described here. Once again, the maximum amplitude is 1h, representing one gyro’s worth of angular but furthermore, there is an almost noticeable pattern in the results. Iterations 201, 301, 401, 501, 601, momentum, but furthermore, there is an almost noticeable pattern in the results. Iterations 201, 301, 701, and 801 all peak at 1h. The skew angle combinations responsible for these peaks can be seen 401, 501, 601, 701, and 801 all peak at 1h. The skew angle combinations responsible for these peaks in Table 4. These results indicate that having two gyro’s with a skew angle of zero results in high can be seen in Table 4. These results indicate that having two gyro’s with a skew angle of zero results allowable angular momentum values. Space is at a premium on a spacecraft and this information in high allowable angular momentum values. Space is at a premium on a spacecraft and this could prove useful if a certain gyro conﬁguration can be made to ﬁt onboard. information could prove useful if a certain gyro configuration can be made to fit onboard. Table 4. Mixed skew angle selected test cases (from 1000 iterations seeking max. singularity Table 4. Mixed skew angle selected test cases (from 1000 iterations seeking max. singularity free free momentum). momentum). Iteration 1 2 3 Iteration β1 β2 β3 201 20 0 0 201 20 0 0 301 30 0 0 301 30 0 0 401 40 0 0 501 50 0 0 401 40 0 0 601 60 0 0 501 50 0 0 701 70 0 0 601 60 0 0 801 80 0 0 701 70 0 0 Skew angles measured in degrees. 801 80 0 0 Skew angles measured in degrees. Afterwards, we iterated both gyros #1 and 2 while holding gyro #3 at a very low value of one degree, Afterwards, we iterated both gyros #1 and 2 while holding gyro #3 at a very low value of one and the results are provided in Table 5, followed by numerous further iterations in Tables 6 and 7. degree, and the results are provided in Table 5, followed by numerous further iterations in Tables 6 and 7. Aerospace 2019, 6, 98 11 of 14 Table 5. Singularity-Free Maximum Angular Momentum Values for Mixed Skew Angles. Angular Momentum 1 2 3 1 90 90 free 0.97 80 1 1 0.93 70 3 1 0.89 60 7 1 0.83 50 12 1 0.79 40 18 1 0.76 30 26 1 0.78 20 38 1 0.85 10 54 1 1.00 1 90 1 Skew angles measured in degrees. Table 6. Singularity-free maximum angular momentum values for various mixed skew angles. 1 2 3 15 30 60 0.54 30 15 60 0.51 30 60 15 0.16 15 60 30 0.16 60 15 30 0.51 60 30 15 0.54 20 40 80 0.34 20 80 40 0.17 40 20 80 0.33 40 80 20 0.17 80 20 40 0.33 80 40 20 0.53 30 60 90 0.12 30 90 60 0.41 60 30 90 0.51 60 90 30 0.41 90 30 60 0.51 90 60 30 0.25 0 45 90 0.73 0 90 45 0.22 45 0 90 0.41 45 90 0 0.22 90 0 45 0.41 90 45 0 0.73 0 30 60 0.52 0 60 30 0.39 30 0 60 0.52 30 60 0 0.39 60 0 30 0.52 60 30 0 0.53 Skew angles measured in degrees. Aerospace 2019, 6, 98 12 of 14 Table 7. Signiﬁcant skew angles and maximum (saturation) angular momentum. ! ! Maximum Momentum, H 1 2 3 0 ! 0 ! f ree 3 0 ! f ree ! 0 3 30 ! 0 ! f ree 3 60 ! 0 ! f ree 3 90 ! 0 ! f ree 3 90 ! f ree ! 90 3 0 ! 90 ! 90 2.449 30 ! 90 ! 60 2.449 60 ! 90 ! 30 2.449 90 ! 90 ! 0 2.449 Skew angles measured in degrees. 4. Discussion Previously established in the 1990’s, control moment gyroscope installation geometry is ubiquitously accepted to include minimally redundant (four) gyros mounted at symmetric skew angles of 56.73 degrees to generate maximum spherical momentum at the outer saturation surface. This benchmark has necessitated considerably voluminous research into singularity avoidance, singularity escape (when avoidance fails), and singularity penetration. In the 2000’s, the benchmark was replaced by the “box-90” conﬁguration prevalent in industry today, following a re-optimization of the symmetric skew angle to maximize the inner zone of singularity-free momentum space. Coincidently, initial asymmetric geometries were introduced without expansive explanatory research, and these geometries were demonstrated to provide useful options for singularity-free momentum generation capabilities in instances where gyros could not all be installed at ninety degrees. With this auspicious beginning, this manuscript provides a more comprehensive treatment of asymmetric options for gyro installations by providing several dozen discrete options for skew angle combinations. Furthermore, surface plots are provided to allow readers to mentally iterate in between the discrete cases provided. This manuscript establishes a reference article to future systems. Future research will include experimental validation of these results on actual space hardware, either in-orbit or using free-ﬂoating spacecraft simulators in laboratories. This study concludes that inﬁnitely more useful asymmetric options are available than currently reﬂected in the predominant literature. No speciﬁc, single conﬁguration is urged; rather, many optional conﬁgurations are identiﬁed as having identical performance characteristics, while occupying a reduced space. By “performance”, we mean both the maximum momentum capability and the maximum singularity-free momentum capability. For engineers utilizing control moment gyroscopes in space-constrained environments, we recommend using the results here to choose options that require less physical space onboard the vehicle than the ubiquitous symmetric array skewed at ninety degrees. 5. Patents Patent 9,567,112 February 14, 2017 by the author illustrates methods of singularity avoidance without null motion and singularity penetration in the event that trajectories enter singularities [47]. Author Contributions: Conceptualization, T.S.; methodology, T.S.; software, T.S., J.L, K.B., J.T.E., Z.L., and E.C.; validation, T.S.; formal analysis, T.S., J.L, K.B., J.T.E., Z.L., and E.C.; investigation, T.S., J.L, K.B., J.T.E., Z.L., and E.C.; writing—original draft preparation, T.S.; writing—review and editing, T.S.; visualization, T.S., J.L, K.B., J.T.E., Z.L., and E.C.; funding acquisition, T.S. Funding: This research received no external funding. The APC was funded by T.S. Acknowledgments: No funds were received to support this research or for covering the costs to publish in open access. The authors give special thanks to Marcello Romano for permitting us to use his course while on sabbatical as the venue for this investigation. Timothy Sands is not a tenured faculty member of Columbia University; rather he is the Associate Dean of the Naval Postgraduate School’s Graduate School of Engineering and Applied Sciences. In order to avoid legal jeopardy, Timothy Sands publishes government-funded research under his association Aerospace 2019, 6, 98 13 of 14 with the Naval Postgraduate School, while publishing non-government funded research under his continuing associations with Stanford and Columbia Universities. Conﬂicts of Interest: The authors declare no conﬂicts of interest. References 1. Euler, L. 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