**1. Introduction**

The discipline of kinematics in its current form has a lengthy history that hark back to at least 1775 with Euler's formulations [1], with almost immediate expansion throughout the nineteenth [2–4] and twentieth centuries [5–29]. There was a particular renaissance in the late twentieth century accompanying the race between the then-Soviet Union and the United States to spaceflight, and its accompanying application toward nuclear deterrence, where considerable lessons from that period (both technical and non-technical) have been expressed in subsequent literature [30–62]. From this distinguished lineage, terminology has converged to refer to sequential rotation sequences (e.g., xyz or 123); which are called *aerospace sequences* about non-repeating axes (also referred to as "Tait-Bryan angles"), while the *orbit sequences* have an axis repeated in the rotation sequence (e.g., xyx or 121, also referred to as "proper Euler angles"), [63]. One non-repeating sequence in particular (commonly called either a 321 or 123 sequence) has become the ubiquitous aerospace sequence. These cited manuscripts substantiate specific technical applications of the orbital and aerospace sequences,

and those technical applications are the focus of this research in hopes of improved performance. With the rise of digital computation in the Information Age, this research critically evaluates the options (seeking diverging truths for the modern times) by addressing such questions as: Is the ubiquitous aerospace sequence (123 or alternatively 321) the best rotation sequence? Evaluation will be driven by two figures of merit: (1) mean and standard deviations of errors indicating how well each rotation sequence represents true roll, pitch, and yaw angles, and (2) computation time to reveals relative numerical superiority in the context of digital computers of the current state of the art. Analysis and results demonstrate the fact that 321 and 123 rotation sequences result in disparate errors and computation time, with the former being relatively superior. Furthermore, the 123 rotation was significantly slower than all the other rotations. Secondly, the symmetric rotations were on average slower than the non-symmetric rotations, despite the same mathematical process and number of steps to solve for the Euler Angles. Lastly, the fastest non-symmetric rotation was the 321 and the fastest symmetric was the 232, slightly faster than the 121 rotation. Taking all Direction Cosine Matrices (DCM) rotations into account, the 232 rotation was the fastest.

The significance of this research cannot be overstated. The current state of the art uses rotational sequences borne from a different era under a different paradigm, but the success of spaceflight has solidified those older results into the current psyche. This manuscript illustrates that improved errors and computational speed are both possible; and in keeping with the acceptance of the older paradigm by evolution of spaceflight, the context of this research is rotational mechanics [62] applied to spacecraft attitude control systems. These advancements complement advanced algorithms [37–45] for nonlinear adaptive system identification [55–59] and control [46–54] permitting improved performance of space missions [35,36,60] in a time when the United States has a pre-occupation with low-end conflicts in the middle east amidst an increasing belligerent world of threats [30–34]. This realization culminated in the recent edict to create a new military service in the US. purely dedicated to space [61].

#### **2. Materials and Methods**

The goal of a spacecraft's Attitude Control System (ACS) is to have a functional system that can move to and hold a specific orientation in three dimensional space, relative to an inertial frame. With regard to classical and rigid body mechanics, the ACS takes into account the Kinetics, Kinematics, Orbital Frame, and Disturbances to control this motion. Figure 1 depicts this process and details the computational steps from desired angle inputs to Euler Angle outputs in the sequence of inputs (from the white blocks in Figure 1) through light grey calculations to dark grey outputs: *ϕ*, *θ*, and *ψ*. Section 2 will explain the theory behind this control system, Section 3 will detail the experimental setup, and Section 4 will show the results and analysis.

**Figure 1.** Overall technical roadmap of the overall process: Euler Angle for Euler's Moment Equation driven by a trajectory-fed feedforward controller.

#### *2.1. Theory of Dynamics*

Dynamics is synonymous with Mechanics. Newton called dynamics the science of machines which may be divided into two parts: statics (later called kinematics) and kinetics [2]. Chasle's Theorem articulates how a complete description of motion may be described as a screw displacement comprised of translation in accordance with Newton's Law and rotations in accordance with Euler's Moment Equations *T* = *J* .*ω* + *ω* × *Jω* [6], where [*J*] is a matrix of mass moments of inertia explained by Kane [23]. Investigation of motion without consideration of the nature of the body that is moved or how the motion is produced is called *Phoronomics*, or "the laws of going", or more commonly but less properly kinematics [13], to be elaborated in Section 2.1.2. The rotation maneuver from one position to another is measured from the inertial reference frame or [XI, YI, ZI] to the final position, the body reference frame or [XB, YB, ZB]. For this simulation, a model was created to rotate from orientation A, [XA, YA, ZA] to orientation B, [XB, YB, ZB]. Since the dot product of two unit vectors is the cosine of the angle between them [25], it is referred to in older works as a direction cosine, which may also be used to describe a satellite in an inclined earth orbit [17] or to express the orientation of the perifocal reference frame with respect to the geocentric-equatorial reference frame [26]. Direction angles are the angles between each coordinate axis and the individual components of the vector. The direction cosines are simply the cosines of these angles [28]. The nature of direction cosines matrices is merely to assemble the direction cosines which completely specify the relative orientation of two coordinate systems [18], thus their appeal as universally applicable tools of kinematics.
