Gimbal Axis Research Articles

Attitude Dynamics/Control of a Dual-BodySpacecraft with Variable-Speed Control Moment Gyros

The dynamics equations of a spacecraft consisting of two bodies mutually rotating around a common gimbal axis are derived by the use of the Newton‐Euler approach. One of the bodies contains a cluster of single-gimbal variable-speed control moment gyros. The equations include all of the inertia terms and are written in a general form, valid for any cluster configurations and any number of actuators in the cluster. A guidance algorithm has been developed under the assumtion that the two bodies of the spacecraft are optically coupled telescopes that relay laser signals. The reference maneuver is found by the imposition of the connectivity between the source and the target on the ground. A new nonlinear control law is designed for the spacecraft attitude and joint rotation by the use of Lyapunov’s direct method. An acceleration-based steering law is used for the variable-speed control moment gyros. The analytical results are tested by numerical simulations conducted for both regulation and tracking cases. I. Introduction T HE dynamics and control of multibody spacecraft are a challenging problem because of the complexity of the dynamics equations and the time-varying inertia of the system. The problem becomes even more interesting when gimbaled momentum exchange devices are considered to control attitude. Control moment gyros (CMGs) are unique among attitude control actuators because they can provide high output torque without using expendable fuels and can provide a level of precision and continuity unachievable with jet thrusters. Indeed, CMGs have been used for decades on space stations and on military spacecraft when fast slewing capability and high pointing accuracy were required. The use of CMGs is also currently under consideration for several future civil spacecraft requiring high agility (as in Refs. 1 and 2). A main drawback to the use of CMGs is the presence of singular gimbal-angle configurations at which the CMG cluster is unable to produce the required torque, or, in some cases, any torque at all. 3−5 Many previous studies have considered the problem of the dynamics and control of spacecraft by the use of single-gimbal CMGs.

System Identification of Adaptive Gyroscopic Damper System Using Sensitivity Analysis

System identification techniques with adaptive gyroscopic damper systems were proposed by applying the capability to change the moment of inertia around its gimbal axis, under a concept of self-identification with adaptive structure systems. The techniques to identify variations of modal parameters were proposed by sensitivity analyses of the natural frequency and the frequency response amplitude with respect to the modal parameters. Further technique to identify the modal parameters was proposed by extending the formulation with sensitivity of the frequency response amplitude with respect to the moment of inertia. To demonstrate the feasibility of each technique, numerical examples with cantilevered beam models were treated. The relations between the errors and non-dimensional system parameters, which include variations of modal parameters, the moment of inertia and the excitation frequency, were investigated numerically. The results indicated that the errors were sensitive to the relation of the system parameters and had singular points where the moment of inertia was close to optimum one to vibration suppression, and the excitation frequency was near the resonant frequency of the beam or the gimbal. The errors at the singular points were reduced by adjusting the moment of inertia or considering the excitation frequency.

Flexible Spacecraft Reorientations Using Gimbaled Momentum Wheels

We study the reorientations of flexible spacecraft using momentum exchange devices. A new concise form of the equations of motion for a spacecraft with gimbaled momentum wheels and flexible appendages is presented. The derivation results in a set of vector nonlinear first order differential equations with gimbal torques and spin axis torques as the control inputs. Feedback control laws which result in smooth reorientations are sought with the goal of minimizing structural excitations. We pay special attention to a class of maneuvers wherein the magnitude of the momentum in the wheel cluster is held constant, resulting in a so-called "stationary platform maneuver." The advantage of this maneuver is that the platform angular velocity remains small throughout, thereby reducing the excitation of the appendages.

Singular Direction Avoidance Steering for Control-Moment Gyros

A new form of the equations of motion for a spacecraft with Single Gimbal Control Moment Gyros is developed using a momentum approach. This set of four vector equations describing the rotational motion of the system is of order 2N + 7 where N is the number of CMGs. The control input is an N x I column vector of torques applied to the gimbal axes. A modification to the singularity robust Lyapunov control law presented by Oh and Vadali is examined and compared to their control law. Specifically, the attempt to avoid singular gimbal configurations is abandoned in favor of simply avoiding movement in the singular direction. The singular value decomposition is used to compute a pseudoinverse which prevents large gimbal rate commands near or at actual singularities.

On a New Six Degree of Freedom Modelling Method for Homing Missiles and its Application for Design/Analysis

Development of an accurate six-degree-of-freedom (6-DOF) simulation model for homing missiles incorporating seeker servosystem detailed modelling is reported. A new modelling concept for seeker servosystem simulation, the Newtonian equivalent model (NEM) has been evolved, where body motion coupling is modelled through forces and moments transformed to the seeker. The NEM-based modelling of seeker head and its integration in 6-DOF has been discussed. In the presence of body coupling, the simulation model for seeker tracking or pointing error, gimbal angles and inertial line of sight (LOS) rates as measured by the seeker mounted rate gyros have been obtained through the new modelling method. A novel method of obtaining optimum pitch and yaw LOS rates for proportional navigation (PN) guidance mechanisation is formulated based on synthesised sight line rates measured in the seeker inner gimbal axis. This new method has been validated through simulation studies using the 6-DOF model developed and by comparing the results with those obtained by the conventional method of generating LOS rates for PN guidance. Other important applications of the 6-DOF model discussed are guidance and control design validation/tuning seeker feedforward compensation design tuning of switchover point from open-loop to closed-loop PN guidance. Importance of the detailed 6-DOF simulation model as the ultimate performance evaluation tool for the weapon system in terms of both terminal performance and adequacy of seeker field of view, gimbal angle freedom, etc. has been brought out.

Modeling Gimbal Axis Misalignments and Mirror Center Offset in a Single-Beam Laser Tracking Measurement System

The calibration of robots and machine tools normally requires external pose measuring devices to provide precision cal ibration measurement data. Among the various coordinate measuring techniques, laser tracking systems based on in terferometry offer the potential for continuously producing noninvasive and highly accurate measurements over a large work volume. This article addresses the error modeling is sue for laser tracking coordinate measuring systems, which is crucial in the accuracy enhancement of such systems. Relative distance measurements provided by laser interfer ometers have an extremely high resolution. However, accuracy errors of a coordinate measuring machine based on laser tracking are dominated by geometric errors in the tracking mir ror system. Major geometric error sources include gimbal axis misalignments and mirror center offset. In this article a kine matic model for the single-beam tracker is developed in which a necessary and sufficient number of kinematic parameters are used to represent these two types of error sources for ar bitrary target positions. This model, together with its error model, which is also presented in this article, can be used for design, calibration, and control of single-beam laser tracking measurement systems.

Optimum Design Method for a Passive Gyroscopic Damper. Numerical Optimization under Random Excitation.

The design method for a gyroscopic mechanism for stabilizing torsional vibration is discussed. The passive gyroscopic damper (PGD) has a rotor in the gimbal rotating at a constant angular velocity. The gimbal is driven by a gyroscopic moment induced by the rotation of the main system, and is passively controlled by the rotational spring and the viscous damper around the gimbal axis. The gimbal rotation induces the resisting gyroscopic moment against excitation. The mechanism enables effective vibration control compared with conventional dynamic vibration absorbers, while it has a fairly simple structure and requires no actuators except for that of the rotor rotation. In the paper, the Galerkin method is applied to analyze the stationary response of the PGD subjected to ergodic Gaussian white noise excitation. The PGD shows considerable nonlinearity under strong excitation which causes large amplitude of the gimbal angle. Numerical optimization techiniques are employed to minimize the rms value of the main system angle. The simulation results are compared with those obtained using the linear design ; thus, the optimization effect is demonstrated.

Optimum Design Method for a Passive Gyroscopic Damper. Fixed Points Theory and Minimum Variance Criteria.

The design methods for a gyroscopic mechanism to stabilize torsional vibration are discussed. The passive gyroscopic damper (PGD) has a rotor in the gimbal rotating at a constant angular velocity. The gimbal is driven by a gyroscopic moment induced by the rotation of the main system, and is passively controlled by the torsional spring and the viscous damper around the gimbal axis. The gimbal rotation again induces the resistive gyroscopic moment against the excitation. The mechanism enables effective vibration control compared with conventional dynamic vibration absorbers, while it has a rather simple structure. Design methods for typical design conditions are developed that give the optimal gimbal spring and the optimal gimbal damper for a given rotor and rotor speed. The fixed point theory for the dynamic vibration absorbers is extended and the results are suitable for harmonic excitations. Closed-form solutions of the minimum variance criteria are derived, which minimize the variance of the main system angle under white noise excitation, taking account of the main system damping. The transmissibility from the excitation to the gimbal angle is estimated to guarantee stable operation for various designs.

Analysis of Apparent Spontaneous Yawing for a Ship Under Way

In this paper a mathematical model is presented to explain the phenomenon of yawing oscillations of a ship, when sailing in a mean straight course with a constant mean speed. With this model, periodic yawing can be simulated as a result of simultaneous periodic rolling and pitching, emphatically without the input of any rudder action. Additionally, the study throws a new light on the old question of whether the Cardan suspension of the compass bowl should be with the outer gimbal axes fore and aft or athwartships.

磁気コンパスボウルの動揺に伴うカードの動きについて

There are some causes of indication error on magnetic compasses when the ship rolls or pitches. These are considered which comes from so called heeling errors and the error caused by the direction of gimbal axis. However, observing the movement of a magnetic compass card in a small boat while sailing, we found that the card are deflected by the swing of the bowl. And it is obviously increased with the rolling and pitching. We simulated the bowl motion as that of which is brought like the real rolling and pitching, and the drifted angle was measured. An example is as follows. A drifted angle of 3.6° of the compass card was obtained in the case that an inclination of the up and down line of the bowl is 10° from the vertical line, and the axis of the direction turns around the vertical line through the compass pivot in a speed of one turn per 15 seconds where the horizontal magnetic flux density is 18μT.

Analysis of a double gimbaled reaction wheel spacecraft attitude stabilization system

Three axis attitude stabilization of a satellite using a single spinning reaction wheel mounted on a two degree-of-freedom passively and actively torqued gimbal system is investigated. The passive control is assumed to be provided by a spring-loaded damper mounted on each of the gimbal axes, while active control results from both the wheel acceleration and the torque applied about the gimbal axes. The stability of the uncontrolled and passively controlled systems is investigated analytically. For constant wheel speed the pitch motion is decoupled from the roll-yaw and gimbal motions. Control laws for the roll-yaw motion are developed based on pole clustering and linear optimal control theory. For the pitch motion control laws are obtained based on classical second order system theory. Estimation techniques are applied to the roll-yaw system for the case when the complete state may not be directly observable (in the absence of a fine yaw position sensor).

A 7,500-Ton-Capacity, Shipboard, Completely Gimbaled and Heave-Compensated Platform

A successful deep-ocean system for deploying and retrieving kiloton loads on a pipe string from a working vessel has been developed. This paper describes the design, construction, installation, testing, and at-sea operational experiences of this unique system onboard the Glomar Explorer. Introduction Just a few years ago, technology for performing heavy work in the deep oceans was rather limited. The size and weight of equipment that could be handled and lowered from a surface vessel to great depths was a primary restriction. Oceanographers worked primarily with winch and cable systems with capacities limited to a few tons. Somewhat larger loads could be handled more proficiently from drillships such as the Glomar Challenger. proficiently from drillships such as the Glomar Challenger. Because of these restrictions, subsea equipment was designed to be small and lightweight. Elaborate buoyancy schemes were necessary to accommodate weights of any magnitude. In early 1970, a program was envisioned that would give new life to deep-ocean exploitation. The goal was to develop a capability to handle kiloton loads at oceanic depths. The Glomar Explorer was a result of this goal. This ship was designed to lower loads of several million pounds to depths exceeding 15,000 ft using a lift pipe. pounds to depths exceeding 15,000 ft using a lift pipe. Maximum loads supported by the ship's pipe hoisting system can exceed 15 million lb. To achieve such load capacity, a highly sophisticated pipe-string design was required. This weight-optimized, tapered pipe string is designed for high-tensile loads with very little margin for bending loads. To isolate the lift pipe from bending loads induced by ship motion, the load-supporting platform was gimbaled and heave-compensated. Gimbal System The heave-compensated and gimbaled platform provides stable support for the hoisting system' and suspended load. Its intent is to assure that minimal bending loads are induced by ship motions into the highly stressed lifting pipe. pipe. The gimbal platform consists of an inner (pitch) gimbal ring and an outer (roll) gimbal ring. The outer gimbal ring is supported on the heave-compensator rams forward and aft through support yokes (see Figs. 1 through 3). The heave-compensator rams are, in turn, supported by the A-frame structure that transversely bridges the ship's open moonpool. The connection between the support yokes and the gimbal ring is through two longitudinal pins and large bearings that serve as the gimbal roll pivot pins and large bearings that serve as the gimbal roll pivot axis. Each yoke rides in vertical guide slots in the A-frame structure to contain the gimbal heave (15-ft total stroke). The inner gimbal structure is connected to the outer gimbal at the pitch axis on large bearings port and starboard through pins cantilevered from the inner gimbal (see Figs. 4 through 6). The inner gimbal, which is independent of ship roll, pitch, and heave motions, forms the heart of the Glomar pitch, and heave motions, forms the heart of the Glomar Explorer's stable-platform concept. The upper pair of hoisting cylinders is mounted on the inner gimbal. Below the inner gimbal, a cage structure is hung that serves to support the lower pair of heavy-lift cylinders. The subbase structure that supports the rig floor and pipehandling derrick is also supported on the inner gimbal pipehandling derrick is also supported on the inner gimbal (see Figs. 2 and 3). The gimbal system is capable of accommodating relative motion between the ship and pipe string of plus or minus 7 1/2 ft heave, plus or minus pipe string of plus or minus 7 1/2 ft heave, plus or minus 8 1/2 degrees roll, and plus or minus 5 degrees pitch. JPT P. 439

On particular solutions of the problem of motion of a gyroscope in gimbal mount: PMM vol. 38, n≗4, 1974, pp. 757–760

Particular solutions of equations of motion of a heavy gyroscope in gimbal mount with the outer gimbal axis horizontal was considered in [1, 2]. Particular solutions of this problem are considered below in the case, when the axis of rotation of the outer gimbal is horizontal and the center of gravity of the gyroscope and of the casing are not lying on the axis of symmetry of the gyroscope ellipsoid of inertia but in a plane passing through the axis of symmetry perpendicularly to the axis of rotation of the inner gimbal. The latter solutions also complement each other symmetrically, similarly to those mentioned above.

On space vehicle attitude stabilization by passive control moment gyros

In this paper the basic equations of motion for space vehicles with various passive control moment gyro (CMG) configurations are presented. As the exact equations of motion are nonlinear, linear models or linearized expressions have been established. This is of major importance especially for system design as the linearized equations allow one to determine explicitly the influence of system parameters on performance. It was found that for step and sinusoidal input functions, the errors due to linearization are less than 1%. Special attention was given to friction phenomena, i.e., viscous and Coulomb friction, which arise in connection with the motion of the CMGs about their respective gimbal axes. It was found that friction decisively influences system performance. Coulomb friction, in general, severely restricts the attitude stabilization capability of the CMGs until the space vehicle has attained a certain angular velocity.

On a certain solution of the problem of motion of a gyroscope on gimbals: PMM vol. 34, n≗6, 1970, pp. 1144–1149

The equations of motion of a heavy gyroscope on gimbals are integrated for an arbitrary position of the center of mass of the gyro housing. In 1958 Chetaev [1] investigated the motion of a heavy gyro on gimbals in the case of vertical position of the outer gimbal axis of rotation (output axis). The center of gravity of the housing and gyro was assumed to coincide with the axis the symmetry of the gyro. Chetaev reduced the problem of integrating the equations of motion to quadratures. These quadratures can be readily extended to the case where the gyroscope is acted along the axis of rotation of its housing by a moment of external forces which is an arbitrary integrable function of the angle of nutation. This problem was considered in [2] under certain assumptions concerning the moments of inertia of the system elements and for certain specific initial data.

Gyro Misalignment and Encoder Quantization Effects During Multi-Axis Evaluation of Strapdown Gyros

This paper presents a quantitative discussion of two error sources which affect the accuracy of data comparison between instrument outputs and reference angle computations during multi-axis evaluation of strapdown gyros. The error due to misalignment of gyro input axes is formulated in terms of closed-form solutions for typical relationships between the three table gimbal rotation histories. If the angular positions of the table gimbal axes are measured by shaft encoders, then any measurement has an uncertainty in the range from plus to minus one-half of the quantization level. It is shown that the effect of encoder quantization error on the computer transformed angles in the bodyfixed coordinate system can be divided into two parts: the initial uncertainty of the zero positions of the gimbals before table motion is begun, and all subsequent error contributions during table test. Typical data for gyro misalignment and encoder quantization error effects are presented.

Satellite attitude control using a torqued, 2-axis-gimbaled boom as the actuator.

Heretofore, satellites requiring precision attitude control have been equipped with momentum wheels and gas jets. This combination can be replaced by a commandable gravitygradient boom and considerably smaller wheels. Each of the two boom gimbal axes is equipped with a torquer and a shaft encoder. Satellite attitude control is maintained using an active attitude reference and torquing against the inertia of the boom. Gravity-gradient torques keep the boom anchored in the gravity field and small momentum wheels damp the boom librations. Yaw axis momentum control utilizes the orbital coupling of the roll/ yaw axes. Results of linear analyses, computer simulations, and some hardware tests exemplify the performance of this Commandable Gravity-Gradient System (COGGS) and demonstrate its feasibility. The system can be expected to statically point to any spot on the earth to within dbO.l0 and to track low-altitude satellites to within ±0.5°.

Dynamics of a Gyroscope Having Oblique Gimbal Axes

The effects are considered of obliquity in the suspension axes of a free two-axis gyroscope supported in gimbals. Three principal types of obliquity are considered, namely (1) non-orthogonality of the bearing axes of the outer and inner gimbals, (2) non-orthogonality of the bearing axes of the inner gimbal and the spin axis of the rotor, in a plane Oyz defined by their usual orthogonal directions, and (3) misalignment of the spin axis of the rotor and a principal axis of the inner gimbal, in a direction out of the plane Oyz. Such obliquities or misalignments could occur as a result of manufacturing errors or they might be introduced deliberately in order to obtain particular dynamical characteristics. The free motion of the system is shown to be described by two coupled second-order non-linear equations, which are solved to first and second approximations for the three cases in which misalignment occurs separately. The first approximation determines the nutational vibration and the second the drift rate. Both are shown to be affected by each type of misalignment: the nutation varies in frequency and mode, and the drift rate in magnitude but not in direction. The relations between misalignment of the spin axis and dynamic unbalance of the inner gimbal are also discussed.

Some Theoretical Considerations in the Measurement of Gravity at Sea

summary The theory of operation of spring gravity meters on a stabilized platform in the presence of horizontal and vertical accelerations is considered in the first section of this paper. It is shown that levelling errors with the same period as the horizontal accelerations are very important. The amplitude of such levelling errors should not exceed about 8 seconds of arc when the amplitude of the horizontal accelerations is 5oooomgal. A second effect, which is present even on a perfectly stabilized platform, has been named the “cross-coupling” effect. It is produced by coupling between the vertical and horizontal accelerations and may, for a typical gravity meter, cause an error in excess of 100mgal when the horizontal and vertical accelerations both have amplitudes of 50000 mgal. The second part of the paper is devoted to a discussion of a gravity meter free to swing in gimbals. An expression for the second-order correction is derived taking into account free and forced oscillations of the gimbal system. This expression takes an especially simple form ½gθ2 (θ is the deflection of the gimbal system from the true vertical) when the gravity sensing element is positioned at the correct distance below the gimbal axis.

Damped Free Oscillations of a Gyroscopic System

THE equations of motion of a gyroscopic system consisting of a symmetric rotor of axial moment of inertia C mounted in heavy gimbals are: where n is the constant angular velocity of the rotor about its axis; θ1 and θ2 are small angles denoting rotations about the outer and inner gimbal axes respectively; c 1 and c 2 are the corresponding coefficients of viscous friction; A is the moment of inertia of the entire system about the outer gimbal axis and B that of the inner gimbal plus rotor about the inner gimbal axis. These equations are based on three principal assumptions: (1) the centres of mass of the rotor and of the gimbals lie at the intersection of the gimbal axes, (2) the axis of the rotor is nearly perpendicular to the axis of the outer gimbal, and (3) the axes of rotation are principal axes of inertia. If there is no friction at the gimbal axes, equations (1) define a free, undamped oscillation in which the rotor axis performs a conical whirl about its central position and the gimbals vibrate at frequency p = Cn/√AB.