A two-step formation-keeping control methodology is proposed that includes both attitude and orbital control requirements in the presence of uncertainties. Based on a nominal system model that provides the best assessment of the real-life uncertain environment, a nonlinear controller that satisfies the required attitude and orbital requirements is first developed. This controller allows the nonlinear nominal system to exactly track the desired attitude and orbital requirements without making any linearizations/approximations. In the second step, a new additional set of closed-form additive continuous controllers is developed. These continuous controllers compensate for uncertainties in the physical model of the satellite and in the forces to which it may be subjected. They obviate the problem of chattering. The desired trajectory of the nominal system is used as the tracking signal, and these controllers are based on a generalization of the concept of sliding surfaces. Error bounds on tracking due to the presence of uncertainties are analytically obtained. The resulting closed-form methodology permits the desired attitude and orbital requirements of the nominal system to be met within user-specified bounds in the presence of unknown, but bounded, uncertainties. Numerical results are provided, showing the simplicity and efficacy of the control methodology, and the reliability of the analytically obtained error bounds.
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.
This paper presents a new, simple, and exact solution to the formation keeping of satellites when the relative distance between the satellites is so large that the linearized relative equations of motion no longer hold. We employ a recently proposed approach, the Udwadia-Kalaba approach, which makes it possible to explicitly obtain the desired control function without making any approximations related to the nonlinearities in the underlying dynamics. We use an inertial frame of reference to describe the motion of a satellite and since no approximations are made, the results obtained apply to situations even when the distance between the satellites is arbitrarily large. The paper deals with a projected circular formation, but the methodology in this paper can be applied to any desired configuration or orbital requirements. Numerical simulations confirm the brevity and the accuracy of the analytical solution to the dynamical control problem developed herein.
This paper deals with finding Lagrangians for damped, linear multi-degree-of-freedom systems. New results for such systems are obtained using extensions of the results for single and two degree-of-freedom systems. The solution to the inverse problem for an n-degree-of-freedom linear gyroscopic system is obtained as a special case. Multi-degree-of-freedom systems that commonly arise in linear vibration theory with symmetric mass, damping, and stiffness matrices are similarly handled in a simple manner. Conservation laws for these damped multi-degree-of-freedom systems are found using the Lagrangians obtained and several examples are provided.
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