Relative Motion Problem Research Articles

Design and Optimization Analysis of an Adaptive Knee Exoskeleton

To solve the problem of undesired relative motion of human-machine interaction positions caused by misalignment of the human-machine joints rotation axis of the knee exoskeleton, this study designed an adaptive knee exoskeleton based on a gear-link mechanism (GLM) by considering the human body as a component of the exoskeleton mechanism. Simultaneously, the concept of the wearable area (WA) was proposed, which transformed the operation of aligning the exoskeleton rotation axis with the human knee joint rotation axis into a "face alignment point" in the sagittal plane, reducing the difficulty of aligning the human-machine joint rotation axis. In the kinematic analysis of GLM, the phenomenon of instantaneous movement of the central axis of the human knee joint was considered. Based on the kinematic model, the WA, velocity transfer ratio, and initial position static stiffness of GLM were analyzed. The NSGA-II optimization algorithm was used to optimize the size parameters of GLM, which increased the WA by 18.4%, the average velocity transfer ratio by 4.98%, and the average initial position static stiffness by 6.01%. Finally, the ability of the exoskeleton to absorb movement displacement (MD) was verified through simulation, and the good human-machine kinematic compatibility of the exoskeleton was verified through wearable tests conducted on the initial mechanism principle prototype.

A Full-Body Relative Orbital Motion of Spacecraft Using Dual Tensor Algebra and Dual Quaternions

This paper proposes a new non-linear differential equation for the six degrees of freedom (6-DOF) relative rigid bodies motion. A representation theorem is provided for the 6-DOF differential equation of motion in the arbitrary non-inertial reference frame. The problem of the 6-DOF relative motion of two spacecraft in the specific case of Keplerian confocal orbits is proposed. The result is an analytical method without secular terms and singularities. Tensors dual algebra and dual quaternions play a fundamental role, with the solution representation being the relative problem. Furthermore, the representation theorems for the rotation and translation parts of the 6-DOF relative orbital motion problems are obtained.

Convex Optimization-Based Techniques for Trajectory Design and Control of Nonlinear Systems with Polytopic Range

This paper presents new techniques for the trajectory design and control of nonlinear dynamical systems. The technique uses a convex polytope to bound the range of the nonlinear function and associates with each vertex an auxiliary linear system. Provided controls associated with the linear systems can be generated to satisfy an ordering constraint, the nonlinear control is computable by the interpolation of controls obtained by convex optimization. This theoretical result leads to two numerical approaches for solving the nonlinear constrained problem: one requires solving a single convex optimization problem and the other requires solving a sequence of convex optimization problems. The approaches are applied to two practical problems in aerospace engineering: a constrained relative orbital motion problem and an attitude control problem. The solve times for both problems and approaches are on the order of seconds. It is concluded that these techniques are rigorous and of practical use in solving nonlinear trajectory design and control problems.

Spacecraft Relative Motion Dynamics and Control Using Fundamental Modal Solution Constants

This paper explores expressing the relative state in the close-proximity satellite relative motion problem in terms of fundamental modal solution constants. The nominal uncontrolled relative state can be expressed in terms of a weighted sum of fundamental and geometrically insightful modal motions. These fundamental motions are obtained using the Lyapunov–Floquet theory. In the case that the dynamics are perturbed by the action of a controller or by unmodeled dynamics, the weights on each fundamental solution are allowed to vary as in a variation-of-parameters approach, and in this manner function as state variables. This methodology reveals interesting insights about satellite relative motion and enables elegant control approaches. This approach can be applied in any dynamical environment as long as the chief orbit is periodic, and this is demonstrated with results for relative motion analysis and control in the eccentric Keplerian problem and in the circular restricted three-body problem. Some commentary on the extension of the methodology beyond the periodic chief orbit case is also provided. This is a promising and widely applicable new approach to the close-proximity satellite relative motion problem.

Exact Closed-Form Solutions of the Motion in Non-Inertial Reference Frames, Using the Properties of Lie Groups SO3 and SE3

The paper offers a general symbolic method to study the motion in a non-inertial reference frame. In order to achieve this, we use the algebraic and geometric properties of the Lie group of special orthogonal tensors, SO3, and the Lie group of the rigid body displacements, SE3. We obtain a simplified form of the initial value problem that models the non-inertial motion using a tensor instrument introduced in this paper. Thus, the study of the motion in a non-inertial reference frame is transferred into the study of a classical motion in an inertial reference frame. The applications of this method refer to solving the relative motion problem and deriving the straightforward solution to classical theoretical mechanics problems. The motion in a uniform gravitational force field in a rotating reference frame, the motion of a charged particle in non-stationary electric and magnetic fields, the exact solution of the relative rigid body motion in the non-inertial reference frame are studied. Using this symbolic method in studying the motion in a non-inertial reference frame reduces the number of computations. In addition, it offers, in some essential particular cases, exact closed-form coordinate-free analytical solutions.

Geometric perspectives on fundamental solutions in the linearized satellite relative motion problem

Understanding natural relative motion trajectories is critical to enable fuel-efficient multi-satellite missions operating in complex environments. This paper studies the problem of computing and efficiently parameterizing satellite relative motion solutions for linearization about a closed chief orbit. By identifying the analytic relationship between Lyapunov–Floquet transformations of the relative motion dynamics in different coordinate systems, new means are provided for rapid computation and exploration of the types of close-proximity natural relative motion available in various applications. The approach is demonstrated for the Keplerian relative motion problem with general eccentricities in multiple coordinate representations. The Keplerian assumption enables an analytic approach, leads to new geometric insights, and allows for comparison to prior linearized relative motion solutions.

Mathematical modeling and simulation of nonlinear spacecraft rendezvous and formation flying problems via averaging method

The relative orbital motion problem of a deputy spacecraft with respect to a chief spacecraft is, generally, described using a set of differential equations governing the motion of the spacecraft relative to each other instead of describing their motion, separately, relative to the Earth. In this paper, new time-varying, time periodic cubic approximation model of spacecraft relative motion is developed from the original nonlinear relative equations of motion with the chief in elliptical orbit. Afterwards, averaging method is applied to the cubic model to obtain asymptotic approximations and periodic solutions. The formulation of the averaging solutions using averaging method affords us the opportunity to have a better insight of the relative motion dynamics. From the numerical simulations, the averaging model is in close agreement with the nonlinear model. This can be attributed to the inclusion of cubic nonlinear terms. The model is amenable to the long-term prediction of the behavior of the relative motion and useful for spacecraft formation flying analysis.

Expansion Maps: Designing Relative Trajectories on Quasi-Periodic Orbits

This work establishes a general-purpose method for designing passive spacecraft formations by characterizing the vehicles’ motion on quasi-periodic tori. First, a torus-centric formulation of relative motion is presented. Then, relative distance between spacecraft is examined for two spacecraft on the same torus separated by small angle displacements. The examination motivates the creation of a design methodology that uses a new means of visualizing relative motion on quasi-periodic orbits known as expansion maps. This technique is applied to both the averaged and circular restricted three-body problem (CR3BP). In the CR3BP, both Lissajous and quasi-halo formations are considered. The expansion map demonstrates its capacity to design spacecraft formations in all cases. The creation of this generalized approach toward understanding and designing relative trajectories is a critical next step for formation flight research. Whereas most modern studies solve relative motion problems in particular environments, the techniques discussed in this paper can be applied to a wide variety of systems.

Random Finite Set Theory and Centralized Control of Large Collaborative Swarms

Controlling large swarms of robotic agents presents many challenges, including, but not limited to, computational complexity due to a large number of agents, uncertainty in the functionality of each agent in the swarm, and uncertainty in the swarm’s configuration. This work generalizes the swarm state using random finite set (RFS) theory and solves a centralized control problem with a quasi-Newton optimization through the use of model predictive control (MPC) to overcome the aforementioned challenges. This work uses the RFS formulation to control the distribution of agents assuming an unknown or unspecified number of agents. Computationally efficient solutions are also obtained via the MPC version of the iterative linear quadratic regulator (ILQR), a variant of differential dynamic programming. Information divergence is used to define the distance between the swarm RFS and the desired swarm configuration through the use of the modified distance. Simulation results using MPC and ILQR show that the swarm intensity converges to the desired intensity. Additionally, the RFS control formulation is shown to be very flexible in terms of the number of agents in the swarm and configuration of the desired Gaussian mixtures. Lastly, the ILQR and the Gaussian mixture probability hypothesis density filter are used in conjunction to solve a spacecraft relative motion problem with imperfect information to show the viability of centralized RFS control for this real-world scenario.

Satellite Formation Flying Maneuver Optimal Control

A task of a pair formation flying satellites optimal relative motion control is described. It is presented as a Lagrange problem of satellite relative motion by the criterion of the control acceleration minimization. The сontrol acceleration term corresponds to the term of a fuel flow or a satellite specific impulse. On the basis of a Hill-Clohessy-Wiltshire equation a mathematical model of the relative motion of a pair of satellites is obtained. One satellite is controlled and another is noncontrolled. Analytical description of such relative motion is presented. The optimization criterion considers control acceleration minimization with fixed boundary conditions and a fixed time interval. The system of Euler-Lagrange equations is obtained as a necessary condition for the extremum existence. An analytical solution for the Lagrange problem is obtained. Relative motion simulation for given examples is performed. The example studies relative motion by distance, relative attitude and lateral deviation parametres and four time intervals, corresponding to half orbit length, one, two and four orbit length. The correlation of optimization criterion value and duration of the maneuver is determined. Direct dependence between duration of maneuvers, control acceleration magnitude and control acceleration costs is presented. Correlation between duration of maneuvers and shape of the optimal trajectory is studied. Practical application of this paper results is discussed. An algorithm of a formation flying relative motion control is provided. The algorithm includes stages of an initial relative position definition, the required relative position and duration of a maneuver definition, constants of integration evaluation, optimal control acceleration synthesis.

Bounded relative motions near Keplerian orbits in spherical coordinates

The problem of relative motion for arbitrary Keplerian orbits is modeled in a curvilinear coordinate system. A set of spherical coordinates rather than the classical Cartesian coordinates is used. The techniques such as linearization and normalization are performed to simplify the relative equations of motion as a linear form that has a similar structure as the Tschauner-Hempel (T-H) equations but expressed in spherical coordinates. The analytical solution in spherical coordinates is obtained by using the same technique as shown in the classical T-H equations. Based on this solution, bounded relative motions and the boundedness conditions are recognized for elliptical, parabolic and hyperbolic orbits, respectively. The geometry of bounded relative motions near parabolic and hyperbolic orbits are revealed, which has not been reported in literatures yet. It shows a strong connection with the “leader-follower” formation near elliptical orbits. Numerical simulation demonstrates the improved accuracy of the newly-developed solution and shows the existence of bounded relative trajectories for unbounded reference orbits.

Decentralized fault-tolerant control for multiple electric sail relative motion at artificial Lagrange points

This paper establishes a distributed fault-tolerant control framework for E-sail relative motion at Sun-Earth artificial Lagrange points, capable of accounting for unexpected E-sail actuator fault like effectiveness loss and bias error. The E-sail relative motion problem is formulated on the linearized dynamics, and the steering control is conducted by suitably adjusting the sail attitude and sail lightness number. The proposed control strategy facilitates the E-sail relative distances to evolve within a bounded range and to synchronously converge to the desired values, while collision avoidance of the relative motion is ensured as well. In particular, the locally shared information among the E-sails is assumed to be characterized by their relative distances only, thus extending the existing results where the inter-sail communication capability is static. The concept of distributing multiple E-sails in cluster flight is useful for the deep-space formation missions such as the DARWIN mission, in which a multi-point measurement of the space environment is required. Illustrative examples show the validity of the proposed method in a typical mission scenario.

A Davenport dual angles approach for minimal parameterization of the rigid body displacement and motion

In this paper, the problem of the decomposition of a rigid body displacement by a succession of three rigid body displacements with specified screw-axis is completely solved. Using the isomorphism between the Lie group of the rigid body displacements and the Lie group of the orthogonal dual tensors, the necessary and sufficient conditions for which the decomposition is possible are given. It will be determined either the closed form the dual angles of the screw displacements (translation and rotation). The results are coordinate-free, and they are obtained using only algebraic elements of tensor calculus. In the specific case of a screw axis is perpendicular to the other two, the decomposition is possible for any rigid body displacement. So, a result that generalizes the Davenport decomposition in case of rotation is obtained. A Davenport dual angles decomposition is a new minimal parameterization of the rigid body displacement.For rigid body motion, the kinematic equations that link the instantaneous dual angular velocity to the time variation of Davenport dual angles are deduced. The cases of singularity of the decomposition are identified and a physical interpretation of these cases is given. The results interest the inverse kinematic of robotics, control theory and astrodynamics (full body relative orbital motion problem).

Saturated adaptive sliding mode control for autonomous vessel landing of a quadrotor

An autonomous vessel landing control algorithm of a quadrotor with input saturation and parametric uncertainties is investigated. To facilitate the controller design, the problem of vessel landing is converted from general trajectory tracking problem of a quadrotor to a stabilisation problem of relative motion. A non-linear and coupled six-degrees-of-freedom relative position and attitude model with uncertain parameters and external disturbances is established. The proposed controllers are composed of a relative position controller (RPC) and a relative attitude-altitude controller (RAC). The quadrotor is first commanded by RPC to reach above the vessel, as it reaches, RAC is initiated to guide the quadrotor to descend steadily on the vessel. Both RPC and RAC employ the saturated adaptive sliding mode control technique. The parametric uncertainties and disturbances are estimated by adaptive algorithms, while the control input saturation effect is compensated by linear compensators. All signals in the closed-loop systems are proved uniformly ultimately bounded via Lyapunov theory. Numerical simulations validate the effectiveness of the proposed control approach.

Equinoctial elements for Vinti theory: Generalizations to an oblate spheroidal geometry

Vinti theory constructs orbits on an oblate spheroidal geometry, naturally encoding the gravitational potential of an oblate spheroid in the coordinates. Classical techniques use spherical geometry. Recent work applied Vinti theory to the relative motion problem by way of a linear dynamical model, which is nonsingular in the oblate spheroidal element space. But as with classical spherical elements, the linear mapping between classical spheroidal elements and inertial rectangular coordinates becomes singular for small eccentricities and/or inclinations in the sense of linear dependence of columns in the Jacobian. To mitigate these practical issues, the standard (spherical) equinoctial elements are chosen to inform in a natural way their generalization to a new nonsingular element set: the oblate spheroidal equinoctial orbital elements. The spherical equinoctial elements can be considered a special case of the spheroidal equinoctial elements in the same way that spherical coordinates can be considered a special case of oblate spheroidal coordinates. The new element set is defined and algorithms for converting between spheroidal equinoctial elements and inertial coordinates are derived. Similarities and differences between spheroidal and spherical equinoctial elements are emphasized for clarity, both in terms of the form of equations and geometrical interpretation. The transformations are valid away from the nearly rectilinear orbit regime and are exact except near the poles. When near the poles, the transformations match the accuracy of the approximate analytical solution, which has been developed to the third order in J2 in the literature. As a result, the singularity on the poles is completely eliminated for the first time.

Relative tracking control of constellation satellites considering inter-satellite link

In this article, two main issues related to the large-scale relative motion of satellites in the constellation are investigated to establish the Inter Satellite Link (ISL) which means the dynamic and control problems. In the section related to dynamic problems, a detailed and effective analytical solution is initially provided for the problem of satellite relative motion considering perturbations. The direct geometric method utilizing spherical coordinates is employed to achieve this solution. The evaluation of simulation shows that the solution obtained from the geometric method calculates the relative motion of the satellite with high accuracy. Thus, the proposed analytical solution will be applicable and effective. In the section related to control problems, the relative tracking control system between two satellites will be designed in order to establish a communication link between the satellites utilizing analytical solution for relative motion of satellites with respect to the reference trajectory. Sliding mode control approach is employed to develop the relative tracking control system for body to body and payload to payload tracking control. Efficiency of sliding mode control approach is compared with PID and LQR controllers. Two types of payload to payload tracking control considering with and without payload degree of freedom are designed and suitable one for practical ISL applications is introduced. Also, Fuzzy controller is utilized to eliminate the control input in the sliding mode controller.

The theory of asynchronous relative motion II: universal and regular solutions

Two fully regular and universal solutions to the problem of spacecraft relative motion are derived from the Sperling–Burdet (SB) and the Kustaanheimo–Stiefel (KS) regularizations. There are no singularities in the resulting solutions, and their form is not affected by the type of reference orbit (circular, elliptic, parabolic, or hyperbolic). In addition, the solutions to the problem are given in compact tensorial expressions and directly referred to the initial state vector of the leader spacecraft. The SB and KS formulations introduce a fictitious time by means of the Sundman transformation. Because of using an alternative independent variable, the solutions are built based on the theory of asynchronous relative motion. This technique simplifies the required derivations. Closed-form expressions of the partial derivatives of orbital motion with respect to the initial state are provided explicitly. Numerical experiments show that the performance of a given representation of the dynamics depends strongly on the time transformation, whereas it is virtually independent from the choice of variables to parameterize orbital motion. In the circular and elliptic cases, the linear solutions coincide exactly with the results obtained with the Clohessy–Wiltshire and Yamanaka–Ankersen state-transition matrices. Examples of relative orbits about parabolic and hyperbolic reference orbits are also presented. Finally, the theory of asynchronous relative motion provides a simple mechanism to introduce nonlinearities in the solution, improving its accuracy.

Onboard Exact Solution to the Full-Body Relative Orbital Motion Problem

This paper provides a representation theorem of the onboard exact solution for the six-degree-of-freedom relative orbital motion problem. This problem is quite important, due to its numerous applications: spacecraft formation flying, rendezvous operations, autonomous missions. The common approach is to consider the relative translation and rotation dynamics for the leader–deputy spacecraft formation to be modeled using vector-based formalisms. The proposed approach makes use of a complete dual orthogonal tensor-based construction, which allows for a new description of the full-body relative orbital motion problem. The approach chosen for representing the solution is very short in comparison with the previous reports on the same problem and gives a new formulation that could give important computational benefits.

Invariant Manifold and Bounds of Relative Motion Between Heliocentric Displaced Orbits

This paper discusses a methodology for modeling the relative motion between heliocentric displaced orbits by using the Cartesian state variables in combination with a set of displaced orbital elements. Similar to classical Keplerian orbital elements, the newly defined set of displaced orbital elements has a clear physical meaning and provides an alternative approach to obtain a closed-form solution to the relative motion problem between displaced orbits, without linearizing or solving nonlinear equations. The invariant manifold of relative motion between two arbitrary displaced orbits is determined by coordinate transformations, thus obtaining a straightforward interpretation of the bounds, namely, maximum and minimum relative distances of three directional components. The extreme values of these bounds are then calculated from an analytical viewpoint, both for quasi-periodic orbits in the incommensurable case and periodic orbits in the 1:1 commensurable case. Moreover, in some degenerate cases, the extreme values of relative distance bounds can also be solved analytically. For each case, simulation examples are discussed to validate the correctness of the proposed method.

Trajectory Guidance Using Periodic Relative Orbital Motion

Relative motion of one satellite about another in circular orbit, where the two objects have the same semimajor axis, is periodic in the linearized approximation. A set of orbital elements, the geometric relative orbital elements, which are an exact geometric analogy to the classical orbital elements, can be defined. The relative orbit is manifestly seen to be an ellipse or circle in apocentral coordinates, analogous to perifocal coordinates in inertial motion and different from the local-vertical local-horizontal Cartesian coordinates customarily used for analysis of relative motion problems. These provide a way to do relative motion trajectory design and guidance that stands in contrast to the use of Cartesian coordinates. Algorithms are presented for computing the delta- and timing for impulsive maneuvers to change a single element: two that change the plane and one that changes the size of the relative orbit. The change in size is a two-maneuver transfer and uses the solution of the three-point periodic boundary value problem, also solved and presented here. This solution permits the direct computation, with no iteration or searching, of the periodic relative orbit that connects two points. Optimization to minimize fuel use can be done with a one-dimensional search.