Explicit solution to the full nonlinear problem for satellite formation-keeping

Abstract

a b s t r a c tThis paper presents simple and exact formation-keeping guidance schemes that use a new method that is rooted in some recent advances in analytical dynamics. As a result of this new approach, explicit control inputs to exactly maintain a given formation configuration are easily determined using continuous thrust propulsion systems. The complete nonlinear problem is addressed, and no linearizations and/or approximations are made. The approach provides a marked improvement over existing results in that the control forces, which cause geometric formation-keeping constraints to be exactly satisfied for arbitrary reference orbits, are found in closed form. For Keplerian reference orbits, a much simpler and explicit expression for the control needed to exactly satisfy formation-keeping constraints than hereto available is obtained. The paper also includes explicit control results when the follower is inserted into orbit with incorrect initial conditions, as usually happens in practice. The Hill reference frame, which is often more intuitive for formation-keeping, is used in the analysis. While this paper takes an example of a projected circular formation, the methodology that is developed can be applied to any desired configuration or orbital requirements. Extensive computational simulations are performed to demonstrate the ease of implementation, and the numerical accuracy provided by the approach developed herein.

Dynamic modeling of SCARA robot based on Udwadia–Kalaba theory

“…22 In other words, the initial conditions are incompatible with the constraint equations. It is assumed that the initial configuration for simulation are as follows: Figure 2 represents the dynamic responses at each joint of the SCARA robot subjected to the space trajectory constraints, in which the solid curves and the dash curves represent the numerical value and the theoretical value, respectively.…”

“…Thus, from equation (52), a more general constraint equation is obtained, which nonetheless retains the form of equation (49). 22 Thus, utilizing the Baumgarte stabilization method, the equations of motion for a system subjected to constraints are stated in the form…”

Methodology for Satellite Formation-Keeping in the Presence of System Uncertainties

“…in which the components of the transformation matrix R are given in [21], and r L is the constant distance between the leader satellite and the center of Earth. In Eq.…”

Application of the Udwadia–Kalaba approach to tracking control of mobile robots

“…Inspired by results related to the analytical dynamics with constrained motion, Udwadia creatively proposed a novel methodology for controlling general, nonlinear, structural and mechanical systems [30,31] and used this methodology to solve the control problem of rotational rigid body [32] and satellite formation-keeping [33][34][35]. Using the fundamental equation, Udwadia et al also successfully addressed the control of nonlinear multibody mechanical systems in the presence of system uncertainties [36][37][38][39][40].…”