Eq. 1 illustrates the fundamental relationships of both translational and rotational motion may be expressed as so-called double-integrators, meant to mean the twice integration of the applied force or torques produces the respective translational or rotational displacement. F ¯ = m ¯ ¯ a ¯ = m ¯ x ¯ ¨ = m ¯ d 2 x ¯ d t 2     ↔     T ¯ = J ¯ ¯ α ¯ = J ¯ ¯ ω ¯ ˙ = J ¯ ¯ θ ¯ ¨ = J ¯ ¯ d 2 θ ¯ d t 2 , (1)

Eq. 1 comprises two sets of three equations each for translation and rotation combining for six equations of mechanical motion. For simplicity of expression, states referred to generically as x can represent rotations ( θ ) with regards to the basic, shared motion described by the double-integrator. The transport theorem described next will generate differences in the governing equations for translation and rotation.

Attach three mutually perpendicular unit vectors to each frame: the non-rotating inertial frame and body-fixed frame. The meaning of differentiation of vectors when specification is made of differentiation with respect to a specific frame. Both rotational and translational motion relative to the non-rotating reference frame may be represented by double-integrators in accordance with Eq. 2. m ¯ ¯ a ¯ | r e l a t i v e = m ¯ ¯ x ¨ | r e l a t i v e = m ¯ ¯ d 2 x d t 2 | r e l a t i v e , (2)

Theorem 1. Transport Theorem. The derivative of any vector expressed in the coordinates of a rotating reference frame equals the sum of the derivative relative to a non-rotating reference frame plus the cross-product of the angular velocity and the vector.

Proof of Theorem 1. The proof of this well-known theorem is provided by Kinematics Handout - MIT OpenCourseWare, 2021. The tedious process may be summarized as follows: 1) express the position vector with respect to the non-rotating inertial reference frame; 2) differentiate to find the expression for velocity remembering to differentiate both the component measurements and the unit vectors; 3) simplify and substitute the defined unit vectors, define the angular velocity in the direction perpendicular to the two-dimensional space of rotation; and 4) substitute the newly defined angular velocity to arrive at the transport theorem as expressed in Eq. 3. □ J ¯ ¯ d ω ¯ d t = J ¯ ¯ d ω ¯ d t | r e l a t i v e + ω ¯ × J ω ¯ , (3)

The inclusion of theorem 1, despite being very well-known, is purposely performed to emphasize the most novel proposals presented. In particular, Eqs. 2 and 3 are ubiquitously approximated first by Eq. 1 and also often by linearization of Taylor’s Series of each equation, respectively.

Proportional plus velocity control (Chai et al., 2019) utilizes proportional control by forming a state error scaled by a proportional gain adding a negative gained value of velocity (translational or rotational), as elaborated in Eq. 5. The velocity channel is not a differentiated version of the position or angle channel, as is the case with classical cascaded control topologies of PD, PI, and PID types (proportional plus derivative, proportional plus integral, and proportional plus integral plus derivative, respectively). J ¯ ¯ x ¯ ¨ + K ¯ ¯ V x ¯ ˙ + K ¯ ¯ P x ¯ = K ¯ ¯ P x ¯ d     ↔     u ¯ = K ¯ ¯ p ( x ¯ d − x ¯ ) − K ¯ ¯ v x ¯ ˙ x ¯ ( s ) x ¯ d ( s ) = K ¯ ¯ p J ¯ ¯ s 2 + K ¯ ¯ v s + K ¯ ¯ p → C . E . :   s 2 + K ¯ ¯ v s + K ¯ ¯ p | I = 1 = s 2 + 2 ξ ω n s + ω n 2 . (6)

Gains were tuned for performance specification by equating the ubiquitous closed-loop system Eq. 5 to the performance specified, where C.E. annotates the characteristic equation. The desired rise time established the system natural frequency per t r = 1.8 ω ¯ n , where ω ¯ n ≈ ω ¯ b is the desired control bandwidth; therefore, ω ¯ n = 1.8 t r → K ¯ ¯ p = ω ¯ n 2 . Settling time: oscillation stabilize within 2–5% percent of steady state t s = 4.6 ξ ¯ ¯ ω ¯ n → ξ ¯ ¯ = 4.6 t s ω ¯ n → K ¯ ¯ v = 2 ξ ¯ ¯ ω ¯ n .

Elimination of differentiation in the derivative channel often bestows relative advantage in tracking desired velocity trajectories. Another approach is the optimize gain selection, and this alternative approach is called the linear quadratic regulator.

Eqs. 3 and 4, representing the full, nonlinear, coupled equations of motion in six degrees, may be linearized and be expressed in the form displayed in Eq. 6. This linearization is the basis for the word “linear” in the LQR title. The word “quadratic” refers to selecting gains K that minimize a quadratic cost function displayed in Eq. 7. The LQR solution (Kwakernaak and Sivan, 1972; NASA, 2021a) only bestows optimal solutions for control gains of the form Eq. 8 that minimizes the quadratic cost simultaneously satisfying the (linearized) dynamic constraints displayed in Eq. 5. x ¯ ˙ = A ¯ ¯ x ¯ + B ¯ ¯ u ¯ , (7) J = ∫ 0 ∞ ( x ¯ T Q ¯ ¯ x ¯ + u ¯ T R ¯ ¯ u ¯ ) d t , (8) u ¯ = − K ¯ ¯ x ¯ . (9)

The control designer may select the state weighting matrix Q ¯ ¯ and the control weighting matrix R ¯ ¯ to penalize the state errors and the control effort, respectively. In section 3, equally weighted identity matrices were chosen for both Q ¯ ¯ and R ¯ ¯ . This choice facilitates a multi-faceted comparison in section 3 that does not solely focus on tracking errors or costs. The gains K ¯ ¯ are found using Eq. 9, where the matrix P ¯ ¯ is first found by solving the algebraic relation in Eq. 10, often referred to as a Riccati equation which is most often solved iteratively by a computer (the MATLAB^®/SIMULINK^® lqr command). K ¯ ¯ = R ¯ ¯ − 1 ( B ¯ ¯ T P ¯ ¯ ) (10) A ¯ ¯ T P ¯ ¯ + P ¯ ¯ A ¯ ¯ − P ¯ ¯ B ¯ ¯ R ¯ ¯ − 1 B ¯ ¯ T P ¯ ¯ + Q ¯ ¯ = 0 ¯ (11)

Minimizing a non-quadratic cost function comprised of only the final time (as displayed in Eqs. 11 and 12) constrained with the linearized dynamics of Eq. 6 with costate parameters p(t) used in the Hamiltonian problem formulation leads to time-optimal control (Flugge-Lotz, 1953; Pontryagin et al., 1962; Boltyanskii, 1971; Sands et al., 2009; Sands and Ghadawala, 2011; Duprez et al., 2017; Heidlauf and Cooper, 2017; Baker et al., 2018; Sands, 2019; Smeresky et al., 2020; Arguchintsev and Poplevko, 2021; Malecek, 2021; Srochko et al., 2021). J = ∫ 0 ∞ t f d t , (12) u ¯ = s g n ( < p ( t ) , b i > ) = { 1 − 1 0             i f i f i f < p ( t ) , b i > > 0 < p ( t ) , b i > < 0 < p ( t ) , b i > = 0 . (13)

Simulation subsystems depicted in the appendix execute a bang-bang control where maximal application of control is normalized to unity such that desired unity state and unity time is achieved to aid comparison to the other optimization approaches. One key feature of bang-bang control is the neglecting of the rate end condition leading to a so-called bang-off-bang control, which is not treated here.

Minimizing only the control effort alone (not the state errors) (Sands, 2019) in accordance with Eq. 13 constrained by the double-integrator dynamics of Eq. 2 for specified initial and final conditions permits the solution of a two-point boundary value problem producing optimal control, acceleration, rate, and state profiles displayed in Eq. 14, respectively. Normalization for unity masses or mass moments is included, thus, control and acceleration equations are identical, where non-normalized control may be expressed by scaling the control equation by the masses or mass moments, respectively.

By specifying quiescent initial conditions and using variable scaling and balancing to normalize the final position coordinate to unity, the constants in Eq. 14 may be solved, resulting in Eq. 15, where a = −12, b = 6, and c = d = 0. It should be noted that states are not penalized in the cost function, instead only solution forms that satisfy the boundary values are produced by the two-point boundary value problem from the initial point (x(0),v(0)) = (0,0) to the final point (x(1), v(1)) = (1,0), thus, there is no need to solve an algebraic Riccati equation to produce the optimal control, where an additional benefit of this optimization approach includes the production of optimal state trajectories that will prove useful to decouple the nonlinear coupling effects of the transport theorem described in section 2.2. Scaling and balancing must be performed to normalize the initial and final conditions to zero and unity, and the operations are explained in section 2.11. The solution to the constrained optimization problem listed in Eqs. 2 and 13 was solved analytically and presented recently by Sands (2021) for virtual sensoring, and that solution is presented here in Eqs. 14 and 15. The mathematical development is intentional since 1) the development is well-articulated by Sands (2021) and 2) increased focus on the utilization of these results toward nonlinear equations of motion (presented in Section 2.9). J = 1 2 ∫ 0 ∞ ( u ¯ T u ¯ ) d t , (14) u ¯ ∗ = a ¯ t + b ¯ , v ¯ ˙ ∗ = a ¯ t + b ¯ , v ¯ ∗ = 1 2 a ¯ t 2 + b ¯ t + c ¯ , x ¯ ∗ = 1 6 a ¯ t 3 + 1 2 b ¯ t 2 + c ¯ t + d ¯ , (15) u ¯ ∗ = − 12 ¯ t + 6 ¯ , v ˙ ∗ = − 12 ¯ t + 6 ¯ , v ¯ ∗ = − 6 ¯ t 2 + 6 ¯ t , x ¯ ∗ = − 2 ¯ t 3 + 3 ¯ t 2 . (16)

The open-loop optimal solution embodied in Eqs. 14 and 15 may be updated in real-time using state feedback resulting in real-time optimal control presented in Section 2.7. These optimal states in Eq. 15 are used to form nonlinear controls in Section 2.9.

A corollary is to the open-loop minimum-control optimization in section 2.6 augments the approach with feedback while maintaining the remaining portions of the problem approach. The solution for the constants in between Eqs. 14 and 15 may be accomplished in real-time using feedback but asserting the current position and velocities (translational and rotational) are the initialization points of a new two-point boundary value problem. Eq. (16) may be written in a matrix-vector form as Eq. 17, permitting real-time solution for the integration constants in the vector by inverting the matrix and pre-multiplying both sides of the equation as depicted in (17). Notice the form of the control derived in Eq. 17 is the same as Eq. 14 in section 2.6, where the constants in the optimal solutions are solved in real-time. v ∗ = 1 2 a t 2 + b t + c , x ∗ = 1 6 a t 3 + 1 2 b t 2 & v ∗ ( t f ) = 1 2 a + b = 0 , x ∗ ( t f ) = 1 6 a + 1 2 b = 1 , (17) [ t 0 2 6 t 0 2 2 t 0 1 t 0 2 2 t 0 1 0 1 6 1 2 1 1 1 2 1 1 0 ] ︸ [ T ] { a b c d } ︸ { p } = { x ¯ 0 v ¯ 0 1 0 } ︸ { b } → { a ^ b ^ c ^ d ^ } = [ t 0 2 6 t 0 2 2 t 0 1 t 0 2 2 t 0 1 0 1 6 1 2 1 1 1 2 1 1 0 ] − 1 { x ¯ ( t ) v ¯ ( t ) 1 0 }  and  u ¯ ∗ ≡ a ^ t + b ^ . (18)

One key feature of the open-loop solution method using a two-point boundary-value problem is the enforcement of end conditions producing optimal trajectories for state ( x ¯ *), rate ( v ¯ *), acceleration ( v ¯ ˙ ∗ ), and jerk ( v ¯ ¨ ∗ ), in addition to the formulation of an optimal control, u*. These signals yield the opportunity to formulate decoupling control components to mitigate the transport theorem (illustrated in section 2.9).

Highlighting the matrix inverse in Eq. 17, the possibility of issues inverting a poorly conditioned or rank-deficient matrix may be addressed by monitoring matrix conditioning or determinant and switching away from the feedback solution when encountering rank-deficient instances in favor of the optimal solution in Eq. 15.

As mentioned in Section 2.6, the desire is to use the results of Sections 2.7 , 2.8 applied to nonlinear dynamics coupled by transport theorem. Section 2.2 describes the nonlinear coupling effects of measuring motion in coordinates of rotating reference frames extracted for highlighting in Eq. 18 for translation and rotation, respectively. These effects were neglected when optimizing the double-integrator–based systems of equations or simplified by linearization in other instantiations. Taking advantage of the results in Sections 2.6, 2.7, nonlinear decoupling control components may be formulated using the optimal trajectories as displayed in Eq. 18, where each component (translation and rotation, respectively) should be added to augment the control in Eq. 17 . R o t a t i o n :   u n o n l i n e a r =   ω ¯ × J ¯ ¯ ω ¯ T r a n s l a t i o n : u n o n l i n e a r = m ¯ ¯ d ω ¯ d t × r ¯ + 2 m ¯ ¯ ω ¯ × v ¯ + m ¯ ¯ ω ¯ × ω ¯ × r ¯ }   F e e d b a c k R o t a t i o n :   u n o n l i n e a r = ω ¯ ∗ × J ¯ ¯ ω ¯ ∗ T r a n s l a t i o n :   u n o n l i n e a r = m ¯ ¯ d ω ¯ ∗ d t × r ¯ ∗ + 2 m ¯ ¯ ω ¯ ∗ × v ¯ ∗ + m ¯ ¯ ω ¯ ∗ × ω ¯ ∗ × r ¯ ∗ }   F e e d f o r w a r d , (19)

It is proposed to take the nonlinear control components in Eq. 18 formulated using the optimal trajectories from Eq. 18 and augment them with Eq. 17’s optimal control solutions to comprise the total control for rotation and translation, respectively, displayed in Eq. 19. R o t a t i o n :   u t o t a l ∗ =   a ^ t + b ^ + ω ¯ × J ¯ ¯ ω ¯ T r a n s l a t i o n : u t o t a l ∗ = a ^ t + b ^ + m ¯ ¯ d ω ¯ d t × r ¯ + 2 m ¯ ¯ ω ¯ × v ¯ + m ¯ ¯ ω ¯ × ω ¯ × r ¯ }     F e e d b a c k R o t a t i o n :   u t o t a l ∗ = a ^ t + b ^ + ω ¯ ∗ × J ¯ ¯ ω ¯ ∗ T r a n s l a t i o n :   u t o t a l ∗ = a ^ t + b ^ + m ¯ ¯ d ω ¯ ∗ d t × r ¯ ∗ + 2 m ¯ ¯ ω ¯ ∗ × v ¯ ∗ + m ¯ ¯ ω ¯ ∗ × ω ¯ ∗ × r ¯ ∗ }   F e e d f o r w a r d , (20)

The distinction between feedforward and feedback is determined by the chosen manner of decoupling the nonlinear transport theorem. One option is feedback decoupling, where feedback states are combined (for example, for rotation) as ω ¯ × J ¯ ¯ ω ¯ and added to the optimal control to form a new nonlinear control. On the other hand, feedforward decoupling (used in section 3) combines the optimal equations of state from Eqs. 15 and 16, respectively, combined as ω ¯ ∗ × J ¯ ¯ ω ¯ ∗ and augments the optimal controls with this new nonlinear control. The efficacy of this latter suggestion is evaluated in section 3.

Section 3 validates the proposed developments culminating in the combination of real-time optimal control (Eq. 17) together with transport theorem decoupling in Eq. 19, specifically for rotation using feedforward: u total * = a ^ t + b ^ + ω ∗ × J ω ∗ . After evaluating initial efficacy, effectiveness against realistically varying systems with noise is investigated.

With the inclusion of feedback, noise must be accounted for in the feedback signals (state and rate only here) in the form of random numbers with a standard deviation of 0.01 (1% of the final state value when states are scaled and balanced to unity). In addition, mass and mass moments are assumed to be unknown or not precisely known; therefore, mass and moments were allowed to vary randomly (uniformly) ten percent heavier and lighter. The resulting scatterplots are presented in section 3.

Poorly conditioned problems are those requiring simultaneous mathematical operations on very large and small numbers. A common mitigation strategy is to scale and balance the variables transforming equations to nominally remain of the same order. Scaling problems by common, well-known values permits single developments to be broadly applied to a wide range of state spaces not initially intended. Normalizing time per Eq. 20 restricts simulation time to vary between zero and unity. Scaling mass and mass moments of inertia matrices by their nominal values per Eqs. 21 and 22, respectively, keep their values roughly on the order of unity. Generic displacements (translation or rotation) are normalized in accordance with Eq. 23, where { r } could indicate either general translational or rotational displacement. t ≡ t ¯ t f , (21) m ¯ ¯ ≡ [ m ] [ m ] n o m i n a l , (22) J ¯ ¯ ≡ J J n o m i n a l , (23) r ¯ ≡ { r } { r } f   f o r   t r a n s l a t o i n   a n d   θ ¯ ≡ { θ } { θ } f   f o r   r o t a t i o n . (24)

Double-integrator equations expressing relationships between displacement, displacement rate, and acceleration are canonical relationships used to describe the movement of mass. The relationships are linear, allowing easy control design using classical methods, which predominantly use linear systems methods to design controllers for any equations (including linearizing any nonlinear equations). Each control design technique introduced in section 2 was sequentially used to control the linear, double-integrator system equations, and the results are presented in Figure 3, where Table 9 contains quantitative results corresponding to the qualitative results presented in the multi-plots.

The baseline approach (classical proportional plus velocity, or so-called “P + V” control) tuned to performance specification exhibits better accuracy and lower costs than linear-quadratic optimal regulators of the proportional, derivative (PD) type, but the P + V controller has the highest computational burden as indicated by computational runtime. The embedded differentiation of the noisy feedback signal in the rate channel would logically explain the relatively lower performance of the LQR tracking. Time-optimal (bang-bang) control achieved machine precision state tracking accuracy with the largest rate tracking error of the controllers investigated. Such performance is validated by the instinct that time-optimal control is mathematically designed to achieve the desired state in the shortest time but is not structured to simultaneously achieve rate tracking in minimal time. The cost was the lowest of the controllers investigated, indicating the benefits of not requiring simultaneous rate tracking. Computational runtime was the second largest.

Open-loop optimization calculates the minimum control effort that simultaneously meets state and rate endpoint conditions, and accordingly, both state and rate endpoints are achieved to machine precision, while the computational burden is modest compared to low and high cases. Real-time optimization solves the open-loop optimal control problem in real-time using ideal sensor feedback of state and rate but involves a matrix inverse. Rate and state errors (particularly) are quite small, but machine precision tracking is not achieved. Part of the cause of tracking errors is the inversion of the rank-deficient matrix as the final time is approached. Seeking to ameliorate the issue, switched real-time optimal control is presented where the matrix condition is used to switch away from real-time optimal control to open-loop optimal control during timesteps when matrix inversion becomes poorly conditioned. Machine precision tracking is attained, open-loop optimally low costs are re-achieved, and the computational burden is slightly elevated compared to the best case investigated.

Summarizing the results so far, real-time optimal control (designed only to minimize control effort) with singular switching to counter the deleterious effects of poor matrix conditioning achieves the best simultaneous state and rate error (machine precision) with costs matching the open-loop minimal and average computational burden. Unfortunately, these results are achievable only in idealized circumstances of double-integrators. Expressing motion in coordinates of rotating reference frames introduced nonlinear coupling described in the next section (3.2).

Expressing motion in coordinates of rotating reference frames is referred to as the “transport theorem,” which introduces nonlinear coupling between the six channels of motion that would otherwise have been well-described by simple, linear double-integrators. Very often, linearized system equations or linear assumptions (the double-integrators) are used to design linear controllers. Accordingly, each instance investigated in section 3.1 was applied to nonlinear coupled system equations, including the transport theorem. Increased errors and reduced robustness is generally anticipated since the controllers are not designed to accommodate system nonlinearities specifically. Each control technique introduced in Section 2 was sequentially simulated, and the results are presented in Figure 4, where Table 10 contains quantitative results corresponding to the qualitative results presented in the multi-plots.

It should be noted that all approaches designed to control double-integrators illustrate degraded performance compared to the idealized case investigated in Section 3.1. All methods compared achieved similar costs. The baseline approach (classical proportional plus velocity, or P + V) tuned to performance specification exhibits second-best state accuracy and second-best rate accuracy, while real-time optimal control achieved the lowest state and rate errors but used nearly double the amount of control. No technique achieved machine precision tracking.

Summarizing the results so far, all control techniques are degraded from the idealized case. Real-time optimal control (designed only to minimize control effort) and classical control methods were the most robust, but all three methods utilized substantially more control effort.

Double-integrator relationships (implemented identically as done in sections 3.1 and 3.2) are next augmented with feedback decoupling of the transport theorem using state feedback in Eq. 19. Each control design technique introduced in section 2 was sequentially used to control the nonlinear, double-integrator system equations including transport theorem, and the results are presented in Figure 5, where Table 11 contains quantitative results corresponding to the qualitative results presented in the multi-plots.

All methods are improved by addition of the nonlinear decoupling control designed for the transport theorem. Time-optimal control performs worst regarding state and rate errors, while cost figures are generally increased for methods that effectively track state and rate. Near machine-precision is achieved by open-loop optimal control and switched, real-time optimal control where both are designed to minimize control effort alone (with no state error representation in the minimized cost function). Computational burdens of all approaches are roughly comparable.

Having initially analyzed idealized systems (the double-integrators), nonlinear coupling was induced by the transport theorem with significantly degraded performance using the controllers designed for linear systems. Adding nonlinear control components designed specifically to decouple the transport theorem in feedback roughly restores nominal performances, but feedback remains ideal (without noise). Section 3.4 adds zero-mean Gaussian noise to both sensor types (state and rate)

Double-integrator equations with nonlinearities induced by transport theorem were controlled by linear control designs augmented with nonlinear feedback decoupling designed specifically for transport theorem. Feedback was provided by simulated mixed state and rate sensors, and Gaussian noise was added. Each control design technique introduced in section 2 was sequentially used to control the linear, double-integrator system equations, and the results are presented in Figure 6, where Table 12 contains quantitative results corresponding to the qualitative results presented in the multi-plots.

The baseline approach (classical proportional plus velocity, or P + V) tuned to performance specification exhibits better rate accuracy and lower costs than linear-quadratic optimal regulators of the proportional, derivative (PD) type, but the P + V controller has relatively inferior state errors compared to LQR. Time-optimal control performs poorly in the face of transport theorem and noisy sensors. Real-time optimal control is severely degraded by noise, particularly with respect to rate errors and control effort.

Open-loop optimization and real-time optimal control with singular switching simultaneously achieve the lowest state and rate endpoint errors, with the lowest costs (that may be claimed to meet endpoint conditions), while the computational burden is modest compared to low and high cases.

The claim was just immediately earlier, validating that real-time optimal control with singular switching and transport theorem decoupling was the overall top-performing option, and this section describes the results validating the claim in generally understandable terms (percent performance improvement comparison). Open-loop (control minimizing optimal control constrained to meet end state and rate) performed very well, while real-time optimal control with singular switching matched the performance and was slightly better in terms of computational burden.

The derivation of optimal trajectories (state, rate, acceleration, and jerk) should prove useful in the implementation of deterministic artificial intelligence (Smeresky et al., 2020), which requires some scheme of autonomous trajectory generation. The current state of the art utilizes sinusoidal trajectory generation schemes, and the optimal trajectories illustrated here should have improved efficacy when used to augment deterministic artificial intelligence.