Quadratic Optimal Control in Reorienting a Spacecraft in a Fixed Time Period in a Dynamic Problem Statement

A dynamic problem of a spacecraft (SC) turning from an arbitrary initial angular position to the required final angular position is considered and solved. The termination time of the maneuver is known. To optimize the turn control program, the quadratic criterion of quality is used; the minimized functional characterizes the energy consumption. The construction of the optimal control of the turn is based on quaternion variables and Pontryagin’s maximum principle. It is shown that during the optimal turn, the moment of forces is parallel to a straight line immobile in space and the angular momentum direction in the rotation process of the SC is constant relative to the inertial coordinate system. The formalized equations and calculation expressions to determine the optimal turn program are obtained. For a dynamically symmetric SC, a complete solution to the reorientation problem in the closed form is given. An example and results of the mathematical simulation of the dynamics of the SC’s motion under the optimal control are presented, which demonstrate the practical feasibility of the developed method to control the SC’s attitude.     

Arm path planning of a space robot with angular momentum

Arm path planning of a space robot with angular momentum is considered in this paper. A space robot changes its attitude by the arm motion and angular momentum of the space robot has the possibility to reduce the attitude change. A path planning method of the arm where the final satellite attitude becomes the same as the initial one is proposed. The method derives an approximate path first based on the attitude change when the arm moves along an infinitely small closed curve and then modifies the path by the Newton method. The amplitude of the arm motion decreases with the magnitude of the angular momentum, which shows that the proposed method utilizes the angular momentum effectively.     

On optimal spacecraft damping

The problem of spacecraft damping (damping of initial angular velocity to zero) for a minimal time is studied. Two variants of formulation of the optimization problem are considered; these variants differ in the form of constraints on the control torque. Analytical solution to the formulated problem is obtained in the closed form and numerical expressions for synthesis of optimal angular velocity control program are given. Similar problem of time-optimal angular acceleration of the spacecraft to the given value is also solved. Procedure for determination of the control torque at the initial time instant for the problem of acceleration of the spacecraft to the required angular velocity is presented. Numerical example of solution of the problems of buildup and damping of spacecraft rotation velocity for a minimal time is given.     

基于矢状面质心角动量的人类行走步态周期阶段划分方法

认知人类的步行机理是双足机器人开发的重要基础.在人类行走过程中,外力力矩是影响行走稳定性的决定性因素,步态与外力力矩的相互作用是人类步行机理研究中的关键问题.尽管质心角动量可反映人体受到的外力力矩变化,但会随步态的演化呈现不同的变化规律.以人类自然行走步态为研究目标,通过准确获取人体行走过程中实时运动信息与质心角动量的变化,根据人体行走过程中的外力力矩与质心角动量的角度对人体步态进行力学分析,并结合人体行走过程中的足地关系与矢状面质心角动量变化规律,得出角动量特征点与步态特征点在时间上具有高度一致性的结论,最终实现基于矢状面质心角动量的人类步态周期阶段的精准划分.研究结果对于认知人类步行机理,指导行走康复医疗和双足机器人研发具有重要意义.     

Analytic Controlling Reorientation of a Spacecraft Using a Combined Criterion of Optimality

We consider the problem of optimally controlling the reorientation of a spacecraft (SC) from an arbitrary initial angular state into a given final angular position. We study the case when the minimized functional joins, in the given proportion, the time spent and the integral of the squared modulus of the angular momentum on the reorientation of a SC. The problem is solved in a kinematic setting. We consider two versions of the problem of the optimal rotation of a SC, with bounded and unbounded control. Using the necessary optimality conditions in the form of the Pontryagin maximum principle and the quaternion method for solving control problems on the motion of spacecrafts, we obtain an analytical solution of the posed problem. The solution of the problem is based on the quaternionic differential equation relating the angular momentum vector of a SC with the orientation quaternion of the related coordinate system. We present formalized equations and give computational expressions for constructing the optimal control program. We state the control law as an explicit dependence of the control variables on the phase coordinates. Using the transversality condition as a necessary optimality condition, we determine the maximal value of the modulus of the angular momentum for the optimal motion. For a dynamically symmetric SC, the problem of reorientation in space is solved completely: we obtain the dependences for the optimal law of the change of the angular momentum vector as explicit time functions. We give the results of the mathematical modeling of the motion for optimal control which demonstrate the practical realizability of designed algorithm for controlling the spatial orientation of a SC.     

On the use of free-floating space robots in the presence of angular momentum

Free-floating space manipulator systems include at least one manipulator mounted on an unactuated spacecraft. It is known that such systems exhibit nonholonomic behavior due to angular momentum conservation. In this paper, the initial angular momentum of the system is not assumed to be zero and its influence on system behavior is studied. In contrast to the case of zero initial momentum, in the presence of momentum, the manipulator end effector in general cannot remain at a given location for indefinite time. The paper studies the conditions under which this is possible, rendering the end-effector immune to angular momentum accumulation. The relevant kinematics and dynamics are studied in 2D and 3D systems, and workspace subsets, where the end effector can remain fixed, are identified. Examples illustrate the validity of the results.     

A wideband transmission metasurface for generating Bessel beam carrying orbital angular momentum

Orbital angular momentum (OAM) microwave waves have received increasing research interest because of their potential to improve communication capacity and radar resolution. However, the traditional orbital angular momentum wave has the problem of diffraction divergence. The ideal Bessel beam with orbital angular momentum has the property of nondiffraction. Hence, the Bessel beam carrying orbital angular momentum has great research potential. In this work, a wideband transmission metasurface for generating Bessel beam carrying OAM is designed, fabricated, and measured. And it is the first experimental to show that the Bessel beam carrying OAM has higher gain and lower crosstalk than the traditional OAM wave in far field transmission.     

On the stabilization of a communication satellite without measuring its angular velocity

The motion control about the center of mass of the communication satellite Yamal-200 is considered. Due to possible failure of the angular velocity sensors, an algorithm for maintaining a prescribed attitude without measuring the angular velocity was developed for this satellite. The algorithm ensures the gyroscopic stabilization of the satellite motion about the local vertical. In the course of control, the onboard computer stores projections of the flywheels?? angular momentum vector on the axes of the inertial basis; the control system tries to preserve the angular momentum magnitude and direction in the inertial space based on the measurements of the angular momentum. The workability of the proposed stabilization algorithm was confirmed by flight tests.     

Unloading angular momentum for inertial actuators of a spacecraft in the pitch channel

The problem of gravitational unloading of the angular momentum of inertial actuators of a spacecraft in the pitch channel for circular and elliptic orbits is considered using the band theory of modal control. Control laws for gravitational unloading and stabilization of a given spacecraft position unambiguously determined by the object parameters and given coefficients of characteristic equation are obtained.     

偏置动量卫星阻尼力矩作用下的姿态运动特性分析及控制方法

针对偏置动量卫星阻尼阶段的姿态运动特性进行分析,并基于时域分析方法提出弱偏置角动量的选取方法,改善阻尼过程中滚动-偏航角收敛的动态性能.以实际应用为背景,考虑一箭多星发射模式下,安装约束的限制导致星箭分离后偏置角动量方向严重偏离轨道面法线方向的情况,提出一种基于弱偏置角动量的姿态控制方法,在无需三轴姿态信息的情况下,可以避免由于偏置角动量的定轴性导致滚动-偏航角收敛时间很长的问题.仿真结果表明,所提出的方法能够有效减少姿态捕获时间,且易于实现,具有较高的工程实用价值.     

Parallelizable split-operator propagator for treating Coriolis-coupled quantum dynamics

We show that the split-operator propagator of Feit and Fleck can be converted into a form that is easy to parallelize for a quantum system with Coriolis-coupled quantum dynamics. The angular kinetic energy part of the propagator is replaced by the Crank-Nicholson approximation, which makes the entire propagator tridiagonal in the quantum number Ω (the projection of the total angular momentum quantum number J onto the intermolecular axis). We demonstrate that the modified propagator is numerically stable and accurate, provided that the Crank-Nicholson approximation is applied to the entire angular kinetic energy operator.     

Explicit momentum conserving algorithms for rigid body dynamics

Two new explicit time integration algorithms are presented for solving the equations of motion of rigid body dynamics that identically preserve angular momentum in the absence of applied torques. This is achieved by expressing the equations of motion in conservation form. Both algorithms also eliminate the need for computing the angular acceleration. The first algorithm employs a one-pass predictor-corrector scheme while the second algorithm is based upon the staggered time integration approach of Park. Numerical results are presented comparing the new algorithms to the algorithms of Simo and Wong and Park et al. The predictor-corrector algorithm is shown to suffer weak instabilities while the staggered conserving algorithm exhibits improved performance compared to the staggered algorithm of Park et al.     

A Singularity-free Steering Law of Roof Array of Control Moment Gyros for Agile Spacecraft Maneuver

In this paper, a singularity-free steering law for single gimbal control moment gyros (CMGs) is addressed for agile spacecraft. The geometrical array considered particularly in this work is a roof array due to the simplicity of singularity envelope. A feasible angular momentum chart which can provide a singularity-free bound is employed. The chart allows a guaranteed maximum torque output and angular momentum at any time without concerning the geometrical singularity of the array. Furthermore, a new deterministic allocation algorithm, called half-leading steering logic, of gimbal angular rates, is also suggested instead of the well-known pseudo-inverse technique to meet control torque commands required and to keep away from the singularity. It is noted that a momentum vector recovery to the initial direction is also an important task for the CMG array to overcome the singularity and for the reliable operation of CMGs. An optimization technique is addressed to restore the gimbal vectors back to their original angular position after the attitude reorientation mission. The techniques proposed are demonstrated using illustrative numerical simulations.

     

二维陀螺声子晶体结构力学研究

首先,提出了一种二维平面陀螺声子晶体,得到了不同参数对带隙的影响.该结构由陀螺、外框架、弹簧组成.其次,在结构中只考虑由于陀螺转动引起的角运动,不考虑平移.运用角动量定理,针对该结构建立动力学方程.然后,通过动力学方程,分析其带隙特性得到色散曲面.随后,通过改变结构中的参数,观察带隙的变化,得到了不同参数对带隙的影响.最后,用数值方法验证其带隙的存在,证明了这种二维平面陀螺声子晶体能对扭转振动具有抑制作用.     

Centroidal dynamics of a humanoid robot

The center of mass (CoM) of a humanoid robot occupies a special place in its dynamics. As the location of its effective total mass, and consequently, the point of resultant action of gravity, the CoM is also the point where the robot’s aggregate linear momentum and angular momentum are naturally defined. The overarching purpose of this paper is to refocus our attention to centroidal dynamics: the dynamics of a humanoid robot projected at its CoM. In this paper we specifically study the properties, structure and computation schemes for the centroidal momentum matrix (CMM), which projects the generalized velocities of a humanoid robot to its spatial centroidal momentum. Through a transformation diagram we graphically show the relationship between this matrix and the well-known joint-space inertia matrix. We also introduce the new concept of “average spatial velocity” of the humanoid that encompasses both linear and angular components and results in a novel decomposition of the kinetic energy. Further, we develop a very efficient $O(N)$ O ( N ) algorithm, expressed in a compact form using spatial notation, for computing the CMM, centroidal momentum, centroidal inertia, and average spatial velocity. Finally, as a practical use of centroidal dynamics we show that a momentum-based balance controller that directly employs the CMM can significantly reduce unnecessary trunk bending during balance maintenance against external disturbance.     

Nonlinear control of planar multibody systems in shape space

Modeling, reduction, and nonlinear control of planar multibody systems motivated by the classicalcat-fall problem and the practical problem of reorientation of free-floating multibody satellites with rotational joints using angular-momentum-preserving controls is studied. The system model considered is reduced by the first integral (the system angular momentum) resulting in a Hamiltonian system with a configuration space of relative joint angles (shape space). Reconstruction of dynamics is applied to modify the shape-space model and track the phase shift of the absolute angles. An important reachability result is then proved in the unreduced configuration space. Control synthesis can then be found in a feedback form, solving the reorientation problem completely. Surprisingly, the reachability result breaks down in the case of the planar coupled two-body system with zero angular momentum, proving that the cat-fall phenomenon is definitely nonplanar.     

A special case of spacecraft optimal attitude control

The problem of optimal control of a spacecraft reorientation from an arbitrary initial attitude to a prescribed angular attitude is studied. The reorientation time is given. The case when the quadratic norm of the angular velocity vector of the spacecraft is minimized is studied. Using the necessary optimality conditions in the form of Pontryagin??s maximum principle and the quaternion method for spacecraft motion control, an analytical solution of this problem is obtained. Formal equations are derived and expressions for the optimal control program are obtained. For a dynamically symmetric spacecraft, the angular velocity is obtained in the analytical form. Results of the mathematical simulation of the spacecraft dynamics under the optimal control are presented that demonstrate the practical usefulness of the proposed algorithm for the spacecraft attitude control.     

Controlling space vehicle reorientation with minimal energy integral

We consider the optimal control problem for spatial reorientation of a space vehicle (SV) from an arbitrary initial to a given final angular position. The reorientation time is known in advance, so we are minimizing the rotation energy integral. We present an analytic solution of the proposed problem. With the method of quaternions, we have obtained formal equations and computational expressions for constructing the optimal reorientation program. For a dynamically symmetric SV, we give a complete solution of the SV reorientation problem in closed form. We give an example and mathematical modeling results for SV motion dynamics under optimal control. These results demonstrate that the developed method of controlling SV spatial orientation is feasible in practice.     

Control and simulation of a constrained dynamic two-link system

In this paper a solution to the attitude control problem of a two-link planar system, whose angular momentum about the centre of gravity is constant, is presented. Reference trajectories for simulating the motion of the constrained dynamic system are found by specifying the internal variables of the system and solving for the corresponding external variables. It is shown that the angular momentum constraint has quasi-holonomic character because it can be employed to reduce the dimensionality of the system by eliminating half a degree of freedom. A controller is designed for the reduced dynamic system and tested by digital computer simulation. It is shown that the same controller can be used for the unreduced system if the angular momentum constraint is a linear function of the states of the system. Such an approach, however, cannot be employed if the constraint is a non-linear function of the system state. In this case a controller must be designed which stabilizes both the system and the constraint equation. Computer simulations of the diving motion of a human are presented to illustrate the control approach.