Three Axis Determination from Vector Observations
SINS初始对准算法综述
SI NS初始对准算法综述惯性导航的本质是一个积分航位推算过程,而初始对准的目的就是确定积分初值的过程，包含姿态初值、速度初值和位置初值。
根据对准环境，可将初始对准分为静基座对准、摇摆基座对准和动基座对准。
根据转位方式,又可将静基座对准分为单位置对准、双位置对准和多位置对准，多位置对准可以通过IMU 旋转来改善系统的可观测性，从而提高对准和测漂精度。
一、静基座对准载体在静止或微幅晃动条件下,速度初值为零，位置初值通常由外界输入,所以静基座对准的主要任务在于确走载体姿态。
用于姿态描述的数学方法包括方向余弦矩阵(Direction Cosine Matrix, DCM \欧拉角(Euler Angles \姿态四元数(Quaternion Symmetric Parameters )、旋转矢量(Rotation Vector X 罗德里格参数(Rodrigues Parameters )或吉布斯向量(Gibbs Vector \修正罗德里格参数(Modified Rodrigues Parameters, MRPs )、凯菜■克菜参数(Cayley-Klein Parameters ) 等。
1964年，Stuelpnagel从数学上证明仅采用三维参数表示姿态不可能是全局非奇异的，所以姿态矩阵和四zk元q数表20示16方02法2使2用较为普遍,而欧拉角由于物理意义明确也经常使用。
地球表面可供参考的物理矢量为地球自转角速率ie 和重力加速度g ,只要ie和g不共线(非极区)，就可以通过矢量观测确定最优姿态。
一般认为最优姿态估计算法沿着两条技术路线发展:①确走性算法，即Wahba问题求解[53]；②状态估计法,如卡尔曼滤波方法。
(一)确定性算法1965年Wahba将姿态确定问题描述为带约束的最小二乘估计问题，即Wahba问题。
对于双矢量观测的情况,TRIAD (ThRee-axIs Attitude Determination )方法可以通过直观方式获得闭环解。
三相异步电机最优励磁电流矢量控制
ｏｕｔｐｕｔ ｃｕｒｒｅｎｔ ｉｓ ｅｑｕａｌ，ｔｈｉｓ ｍｅｔｈｏｄ ｃａｎ ｉｎｃｒｅａｓｅ ｔｈｅ ｏｕｔｐｕｔ ｔｏｒｑｕｅ ｏｆｔｈｅ ｍｏｔｏｒ· Ｔｒａｄｉｔｉｏｎａｌ ｖｅｃｔｏｒ ｃｏｎ如ｒ０１ ａｎｄ ｏｐｔｉｍａｌ ｅｘｃｉｔａｔｉｏｎ ｃｕｒｒｅｎｔ ｖｅｃｔｏｒ ｃｏｎｔｒｏｌ ｗｅｒｅ ｓｔｕｄｉｅｄ ｅｘｐｅｒｉｍｅｎｔａｌｌｙ·
ＡＢＳＴＲＡＣＴ……………………………………………………………………………………………………………………ＩＩ
目 录………………………………………………………………………………………………………………………．．ＩＩＩ
第一章绪论…………………………………………………………………………………．１
１．１引言…………………………………………………………………………………………１ １．２现代交流调速技术的发展…………………………………………………………………２
ｏｆ ｍｅ ｍｏｓｔ ｃｒｉｔｉｃａｌ ａｌｇｏｒｉｔｈｍ ｃｏｏｒｄｉｎａｔｅ ｃｏｎｖｅｒｓｉｏｎ ｔｅｃｈｎｏｌｏｇｙ ｉｎ ｖｅｃｔｏｒ ｃｏｎｔｒ０１．ｄｉｓｃｕｓｓｅｓ ｔｈｅ ｄｉ毹ｒｅｎｃｅｓ ｂｅｔｗｅｅｎ ｄｉ佗ｃｔ ｖｅｃｔｏｒ ｃｏｎｔｒｏｌ ａｎｄ ｉｎｄｉｒｅｃｔ ｖｅｃｔｏｒ ｃｏｎｔｒｏｌ，ｆｅｅｄｂａｃｋ，ｆｅｅｄｆｏｒｗａｒｄ ｗｏ ｖｏｌｔａｇｅ ｄｅｃｏｕｐｌｉｎｇ
１．２．１电力电子器件的发展…………………………………………………………………２ １．２．２ ＰＷＭ调制技术………………………………………………………………………．．３ １．２．３矢量控制技术及直接转矩控制技术…………………………………………………３ １．２．４微机控制技术…………………………………………………………………………４ １．３论文主要研究内容及全文结构……………………………………………………………５
数学专业英语课后答案
2.1数学、方程与比例词组翻译1.数学分支branches of mathematics，算数arithmetics，几何学geometry，代数学algebra，三角学trigonometry，高等数学higher mathematics，初等数学elementary mathematics，高等代数higher algebra，数学分析mathematical analysis，函数论function theory，微分方程differential equation2.命题proposition，公理axiom，公设postulate，定义definition，定理theorem，引理lemma，推论deduction3.形form，数number，数字numeral，数值numerical value，图形figure，公式formula，符号notation(symbol)，记法/记号sign，图表chart4.概念conception，相等equality，成立/真true，不成立/不真untrue，等式equation，恒等式identity，条件等式equation of condition，项/术语term，集set，函数function，常数constant，方程equation，线性方程linear equation，二次方程quadratic equation5.运算operation，加法addition，减法subtraction，乘法multiplication，除法division，证明proof，推理deduction，逻辑推理logical deduction6.测量土地to measure land，推导定理to deduce theorems，指定的运算indicated operation，获得结论to obtain the conclusions，占据中心地位to occupy the centric place汉译英（1）数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。
航天器姿态确定(研究现状)
链接地址 /xiaozu/257088?ref=minifeed&sfet=211&fin=1&ff_id=71996187
法优于TRIAD法[3]。此后，Shuster又基于QUEST测量模型证明了：1) Wahba问题 等价于最大似然估计问题[18]，并进一步提出了广义Wahba问题[19]；2) TRIAD法是 一个最大似然估计器[20]； 3)该测量模型的方差阵在EKF公式中可以等效地用非奇异 阵 2 I 33 代替[16]， 该模型也是Shuster教授一生中最引以为自豪的[21]。 针对大视场敏 感器情形，Cheng利用一阶泰勒近似进一步扩展了QUEST测量模型[22]。对于连续 旋转理论， Shuster在文献[23]中正式提出并将该方法应用于解决一般性的姿态奇异 问题，该方法后来在FOAM法[4]、ESOQ2 法[8]中均得到应用。 近年来，虽然没有新的确定性算法出现，但随着Wahba问题本质的探索[19]， 现有算法与最大似然估计关系的揭示[19,20,24]以及方差分析的完善[13]等文献出现， 让 科研工作者对确定性算法有了更深刻的了解，并可进一步掌握方差分析这一有力 工具[25]。 (2) 状态估计法 单纯依靠矢量观测进行姿态解算的确定性方法要求参考矢量足够精确，且易 受敏感器的失准误差、测量误差等因素影响，往往难以满足高精度的定姿要求。 与这类方法相反，状态估计法中的状态量并不仅限于姿态参数，还包括矢量观测 中的一些不确定性参数；另外，现代航天器上的姿态确定系统往往采用多个姿态 敏感器进行组合测量，由于不同敏感器在测量精度、数据更新率上具有较大差异， 一般也需要采用状态估计法进行信息融合。根据姿态角速度信息的获取方式可将 姿态确定方案分为有陀螺方案和无陀螺方案，前者的姿态角速度由速率积分陀螺 测量得到，而后者的姿态角速度一般通过姿态动力学传播得到。 常用的姿态描述参数有方向余弦阵(Direction Cosine Matrix, DCM)、欧拉角 (Euler Angles)、旋转矢量(Rotation Vector)、姿态四元数(Quaternion)或欧拉对称参 数(Euler Symmetric Parameters)、罗德里格参数(Rodrigues Parameters)或吉布斯向量 (Gibbs Vector)、修正罗德里格参数(Modified Rodrigues Parameters, MRPs)、凯莱克莱参数(Cayley-Klein Parameters)等，目前航天器上最常用的姿态参数是四元数， 其优点主要在于用其表示的姿态运动学方程为线性形式，计算量小，且不存在奇 异性。在 1964 年，Stuelpnagel从数学上证明了三维参数用来表示姿态不可能是全 局且非奇异的[26]，因此，虽然旋转矢量[27]、MRPs[28]作为姿态描述参数也有一定应 用，但就描述航天器姿态而言始终不如四元数流行。不过，在航姿系统中常采用 旋转矢量进行快速姿态解算[29]，而欧拉角由于其明显的物理意义也常被用于描述 火箭或导弹的姿态，至于欧拉运动学方程中的奇异问题，可采用双欧拉角法进行 有效解决。另外，文献[30]对姿态描述参数及其运动学方程进行了系统的综述。 扩展卡尔曼滤波(extended Kalman filter, EKF)技术[31-34]常被用于航天器实时姿 态确定，根据姿态参数的选取不同和观测量的不同形式，常见的实现方式有乘性 扩展卡尔曼滤波[34,35](multiplicative ex-tended Kalman filter, MEKF)和加性扩展卡尔
机器人学运动学分析_Kinematics
Dynamic Analysis of Mechanical Systems
Kinematics
Joints can be classified by the degrees of freedom allowed or restricted. The commonly used joints are listed in Table 1. 3.2.3 Mobility Analysis One of the basic steps in the kinematic analysis of mechanical systems is to determine the number of degrees of freedom or independent coordinates required to determine the configuration of the system. It can be shown that the number of constraints provided by a joint is equal to the number of degrees of freedom eliminated from the unconstraint system as a result of using this joint. In the planar system, the configuration of a rigid body undergoing unconstrained motion can be described by three independent coordinates due to the common constraints. Therefore, a planar system with n unconstrained movable bodies has 3 n DOF. If the system has p joints, each of which provides two constraints (note the common constraints), then the number of degrees of the system can be evaluated by the mobility (Glubler) criterion
理论力学英文版试卷B(附参考答案和评分标准)
XXX工业大学2017 ～2018 学年第二学期期末考试试卷学院_________________ 班级__________ 姓名__________ 学号___________ 《理论力学》课程试卷B（附参考答案和评分标准）Instructor: Lin Zongxi题号 1 2 3 4 5 6 总得分题分20 15 15 15 15 20得分1．Choose the correct answers with proper justification (10×2=20, 20%)1 2 3 4 5 6 7 8 9 10C B A B B A C CD C(1) Which one of the following is a scalar quantity? .A) Force B) Position C) Mass D) Velocity(2) If a particle starts from rest and accelerates according to the graph shown, the particle’svelocity at t = 20 s is ____ ______.A) 200 m/sB) 100 m/sC) 0D) 20 m/s(3) A particle has an initial velocity of 3 m/s to the left at s0 = 0 m. Determine its position whent= 3 s if the acceleration is 2 m/s2 to the right. .A) 0 m B) 6.0 m C) 18.0 m D) 9.0 m(4) The directions of the tangential acceleration and velocity are always .A) perpendicular to each other. B) collinear.C) in the same direction. D) in opposite directions.(5) Calculate the impulse due to the force F. .A) 20 kg·m/sB) 10 kg·m/sC) 5 N·sD) 15 N·s(6) As shown in the right figure, two blocks are interconnected by a cable. Which of thefollowing is correct? .A) v A= - v BB) (v x)A= - (v x)BC) (v y)A= - (v y)BD) All of the above.(7) If a particle moves in the x-y plane, its angular momentum vector is in the .A) x direction. B) y direction. C) z direction. D) x-y direction.(8) If the position, s, is given as a function of angular position, θ, by s=10 sin2θ, the velocity, v,is . (Noting that θ=ωt, ω is the angular velocity at time t).A) 20 cos 2θB) 20 sin 2θC) 20 ω cos 2θD) 20 ω sin 2θ(9) If v A=10 m/s, determine the angular acceleration,α, of the rod whenθ=30︒. .A) 0 rad/s2B) -50.2 rad/s2C) -112 rad/s2D) -173 rad/s2(10) A slender rod (mass =M) is at rest. If a bullet (mass = m) is fired with a velocity of v b, theangular momentum of the bullet about A just before impact is .A) 0.5 m v b2B) m v bC) 0.5 m v bD) zeroDetermine the magnitude of the resultant force acting on the screw eye and its direction measured clockwise from the x axis.Solution:Using the vector analysis, we have the x- and y-components of forces 6kN and 2kN as()()()()o1o1o2o26kN cos603kN6kN sin6033 5.196kN2kN cos452 1.414kN2kN sin452 1.414kNxyxyFFFF=-=-=========(4%)Hence, the force vectors are obtained as1112223 5.1961.414 1.414x yx yF FF F=+=-+=+=+F i j i jF i j i j(4%)According to the vector addition rule, the resultant force is expressed in the vector form()()1212121.586 6.61R x x y yRx RyF F F FF F=+=+++=+=-+F F F i ji j i j(3%)So, the magnitude and direction of the resultant force are obtained as()()()()2222o1.586 6.61 6.80kN6.61arccos arccos103.496.80R Rx RyRyRF F FFFθ=+=-+=⎛⎫⎛⎫===⎪ ⎪⎝⎭⎝⎭(4%)Determine the magnitude of the moment of the 200N force about the x axis. Solve the problem using both a scalar and a vector analysis.Solution:(1) Scalar analysis.The x, y and z-components of the force F can be expressed as()()()ooocos200N cos120100Ncos200N cos60100Ncos200N cos45173.2NxyzF FF FF Fαβγ===-======(3%)Thus, the magnitude of the moment of the force F about the x axis is obtained as()()()()23100N0.25m173.2N0.3m17.43N mx y zM F d F d=-+=-+=(4%)(2) Vector analysis.First, we establish a position vector from origin O to point A on the force line of action:0.30.25OA A==+r r j k(2%)And the vector of force F is100100173.2x y zF F F=++=-++F i j k i j k(2%)Hence, the moment of force F about point O yields00.30.2517.432530100100173.2O==-+-i j kM i j k(2%)And the magnitude ofOM about the x-axis is obtained as()17.432530N m17.43N mx OM==-+=i M i i j k(2%)The overhanging beam is supported by a pin at A and the two-force strut BC. Determine the horizontal and vertical components of reaction at A and the reaction at B on the beam.Solution:(1) Free-body diagram (6%)We draw a free-body diagram for the overhanging beam as below:(2) Equations of equilibrium (6%)Consider the counterclockwise moment of the force positive. According to the free-body diagram, we have x- and y-component equations and moment about point A of equilibrium as follows()()()()()()()20,02.51.50,600N800N02.51.50,600N1m2m800N4m900N m02.5x Ax By Ay BA BF F FF F FM F=-==+--==-+--=∑∑∑After solving, we have (3%)3133.33N,950N,3916.67NAx Ay BF F F==-=The negetive sign indicates that the direction sence is oppisite to that shown in the free-body diagram.A 50-lb bar is rotating downward at 2 rad/s. The spring has an un-stretched length of 2 ft and a spring constant of 12 lb/ft. Determine the angle (measured down from the horizontal) to which the bar rotates before it stops its initial downward movement. (The acceleration of gravity g=32.2 ft/s2).(1%) Solution:Potential Energy:Let’s put the datum in line with the rod when θ= 0. Then, at position 1, the gravitational potential energy is zero and the elastic potential energy will beV1 = ½ k (s1)2= ½ (12) (4 - 2)2(2%)Gravitational potential energy at position 2: - (50) (3 sin θ) (2%)Elastic potential energy at position 2: (2%)½ (12) {4 + (6 sin θ) - 2}2 . So, V2 = - (50) (3sin θ) + ½ (12) {4 + (6 sin θ) - 2}2Kinetic Energy:At position 1 (when θ = 0), the rod has a rotational motion about point A.T1 = ½ I A (w2 ) = ½{1/3 (50/32.2) 62} (22 ) (2%)At position 2, the rod momentarily has no translation or rotation since the rod comes to rest. Therefore, T2 = 0. (2%)Now, substitute into the conservation of energy equation.T1 + V1 = T2 +V2 (2%)½{1/3 (50/32.2) 62}( 22 ) + ½ (12) (4 - 2)2= 0.0 - (50)(3 sin θ) + ½(12){4 + (6 sin θ) - 2}2 Solving for sin θ yields (2%)sin θ = 0.4295. Thus, θ= 25.4 deg.6. (20%)The disk is rotating with ω =3 rad/s, α = 8 rad/s 2 at this instant. Determine the acceleration at point B , and the angular velocity and acceleration of link AB .(5%)Solution:Draw the kinematic diagram and then apply the relative-acceleration equation:2//B A AB B A AB B A ω=+⨯-a a αr r (4%)where()()()()222o o /o o o o0.2*3 1.8m/s0.2*8 1.6m/s 0.4cos300.4sin 30,01.6 1.80.4cos300.4sin 301.60.4sin 30 1.80.4cos30n A t A B A AB AB AB B AB ABABa αωααα====↓=-===-+⨯-=++-+a a r i j αk i i j k i j i j(5%)Equating the i and j components, we haveo o 1.60.4sin300 1.80.4cos30B AB AB a αα=+=-+ (2%)After solving, we have222.61m/s 5.2rad/s (counterclockwise)B AB a α=→= (4%)。
控制科学与工程专业词汇(Word最新版)
限制科学与工程专业词汇通过整理的限制科学与工程专业词汇相关文档，渴望对大家有所扶植，感谢观看！objective function 目标函数observability index 可观测指数observable canonical form 可观测规范型on-line assistance 在线扶植on-off control 通断限制open loop pole 开环极点operational research model 运筹学模型optic fiber tachometer 光纤式转速表optimal trajectory 最优轨迹optimization technique 最优化技术orbital rendezvous 轨道交会orbit gyrocompass 轨道陀螺罗盘orbit perturbation 轨道摄动order parameter 序参数orientation control 定向限制originator 始发站oscillating period 振荡周期output prediction method 输出预估法oval wheel flowmeter 椭圆齿轮番量计overall design 总体设计overdamping 过阻尼overlapping decomposition 交叠分解Pade approximation 帕德近似Pareto optimality 帕雷托最优性passive attitude stabilization 被动姿态稳定path repeatability 路径可重复性pattern primitive 模式基元PR (pattern recognition) 模式识别P control 比例限制器peak time 峰值时间penalty function method 罚函数法perceptron 感知器periodic duty 周期工作制perturbation theory 摄动理论pessimistic value 悲观值phase locus 相轨迹phase trajectory 相轨迹phase lead 相位超前photoelectric tachometric transducer 光电式转速传感器phrase-structure grammar 短句结构文法physical symbol system 物理符号系统piezoelectric force transducer 压电式力传感器playback robot 示教再现式机器人PLC (programmable logic controller) 可编程序逻辑限制器plug braking 反接制动plug valve 旋塞阀pneumatic actuator 气动执行机构point-to-point control 点位限制polar robot 极坐标型机器人pole assignment 极点配置pole-zero cancellation 零极点相消polynomial input 多项式输入portfolio theory 投资搭配理论pose overshoot 位姿过调量position measuring instrument 位置测量仪posentiometric displacement transducer 电位器式位移传感器positive feedback 正反馈power system automation 电力系统自动化predicate logic 谓词逻辑pressure gauge with electric contact 电接点压力表pressure transmitter 压力变送器price coordination 价格协调primal coordination 主协调primary frequency zone 主频区PCA (principal component analysis) 主成分分析法principle of turnpike 大道原理priority 优先级process-oriented simulation 面对过程的仿真production budget 生产预算production rule 产生式规则profit forecast 利润预料PERT (program evaluation and review technique) 支配评审技术program set station 程序设定操作器proportional control 比例限制proportional plus derivative controller 比例微分限制器protocol engineering 协议工程prototype 原型pseudo random sequence 伪随机序列pseudo-rate-increment control 伪速率增量限制pulse duration 脉冲持续时间pulse frequency modulation control system 脉冲调频限制系统pulse width modulation control system 脉冲调宽限制系统PWM inverter 脉宽调制逆变器pushdown automaton 下推自动机QC (quality control) 质量管理quadratic performance index 二次型性能指标qualitative physical model 定性物理模型quantized noise 量化噪声quasilinear characteristics 准线性特性queuing theory 排队论radio frequency sensor 射频敏感器ramp function 斜坡函数random disturbance 随机扰动random process 随机过程rate integrating gyro 速率积分陀螺ratio station 比值操作器reachability 可达性reaction wheel control 反作用轮限制realizability 可实现性,能实现性real time telemetry 实时遥测receptive field 感受野rectangular robot 直角坐标型机器人rectifier 整流器recursive estimation 递推估计reduced order observer 降阶观测器redundant information 冗余信息reentry control 再入限制regenerative braking 回馈制动，再生制动regional planning model 区域规划模型regulating device 调整装载regulation 调整relational algebra 关系代数relay characteristic 继电器特性remote manipulator 遥控操作器remote regulating 遥调remote set point adjuster 远程设定点调整器rendezvous and docking 交会和对接reproducibility 再现性resistance thermometer sensor 热电阻resolution principle 归结原理resource allocation 资源安排response curve 响应曲线return difference matrix 回差矩阵return ratio matrix 回比矩阵reverberation 回响reversible electric drive 可逆电气传动revolute robot 关节型机器人revolution speed transducer 转速传感器rewriting rule 重写规则rigid spacecraft dynamics 刚性航天动力学risk decision 风险分析robotics 机器人学robot programming language 机器人编程语言robust control 鲁棒限制robustness 鲁棒性roll gap measuring instrument 辊缝测量仪root locus 根轨迹roots flowmeter 腰轮番量计rotameter 浮子流量计，转子流量计rotary eccentric plug valve 偏心旋转阀rotary motion valve 角行程阀rotating transformer 旋转变压器Routh approximation method 劳思近似判据routing problem 路径问题sampled-data control system 采样限制系统sampling control system 采样限制系统saturation characteristics 饱和特性scalar Lyapunov function 标量李雅普诺夫函数SCARA (selective compliance assembly robot arm) 平面关节型机器人scenario analysis method 情景分析法scene analysis 物景分析s-domain s域self-operated controller 自力式限制器self-organizing system 自组织系统self-reproducing system 自繁殖系统self-tuning control 自校正限制semantic network 语义网络semi-physical simulation 半实物仿真sensing element 敏感元件sensitivity analysis 灵敏度分析sensory control 感觉限制sequential decomposition 依次分解sequential least squares estimation 序贯最小二乘估计servo control 伺服限制，随动限制servomotor 伺服马达settling time 过渡时间sextant 六分仪short term planning 短期支配short time horizon coordination 短时程协调signal detection and estimation 信号检测和估计signal reconstruction 信号重构similarity 相像性simulated interrupt 仿真中断simulation block diagram 仿真框图simulation experiment 仿真试验simulation velocity 仿真速度simulator 仿真器single axle table 单轴转台single degree of freedom gyro 单自由度陀螺single level process 单级过程single value nonlinearity 单值非线性singular attractor 奇异吸引子singular perturbation 奇异摄动sink 汇点slaved system 受役系统slower-than-real-time simulation 欠实时仿真slow subsystem 慢变子系统socio-cybernetics 社会限制论socioeconomic system 社会经济系统software psychology 软件心理学solar array pointing control 太阳帆板指向限制solenoid valve 电磁阀source 源点specific impulse 比冲speed control system 调速系统spin axis 自旋轴spinner 自旋体stability criterion 稳定性判据stability limit 稳定极限stabilization 镇静，稳定Stackelberg decision theory 施塔克尔贝格决策理论stateequation model 状态方程模型state space description 状态空间描述static characteristics curve 静态特性曲线station accuracy 定点精度stationary random process 平稳随机过程statistical analysis 统计分析statistic pattern recognition 统计模式识别steady state deviation 稳态偏差steady state error coefficient 稳态误差系数step-by-step control 步进限制step function 阶跃函数stepwise refinement 逐步精化stochastic finite automaton 随机有限自动机strain gauge load cell 应变式称重传感器strategic function 策略函数strongly coupled system 强耦合系统subjective probability 主观频率suboptimality 次优性supervised training 监督学习supervisory computer control system 计算机监控系统sustained oscillation 自持振荡swirlmeter 旋进流量计switching point 切换点symbolic processing 符号处理synaptic plasticity 突触可塑性synergetics 协同学syntactic analysis 句法分析system assessment 系统评价systematology 系统学system homomorphism 系统同态system isomorphism 系统同构system engineering 系统工程tachometer 转速表target flow transmitter 靶式流量变送器task cycle 作业周期teaching programming 示教编程telemechanics 远动学telemetering system of frequency division type 频分遥测系统telemetry 遥测teleological system 目的系统teleology 目的论temperature transducer 温度传感器template base 模版库tensiometer 张力计texture 纹理theorem proving 定理证明therapy model 治疗模型thermocouple 热电偶thermometer 温度计thickness meter 厚度计three-axis attitude stabilization 三轴姿态稳定three state controller 三位限制器thrust vector control system 推力矢量限制系统thruster 推力器time constant 时间常数time-invariant system 定常系统，非时变系统time schedule controller 时序限制器time-sharing control 分时限制time-varying parameter 时变参数top-down testing 自上而下测试topological structure 拓扑结构TQC (total quality control) 全面质量管理tracking error 跟踪误差trade-off analysis 权衡分析transfer function matrix 传递函数矩阵transformation grammar 转换文法transient deviation 瞬态偏差transient process 过渡过程transition diagram 转移图transmissible pressure gauge 电远传压力表transmitter 变送器trend analysis 趋势分析triple modulation telemetering system 三重调制遥测系统turbine flowmeter 涡轮番量计Turing machine 图灵机two-time scale system 双时标系统ultrasonic levelmeter 超声物位计unadjustable speed electric drive 非调速电气传动unbiased estimation 无偏估计underdamping 欠阻尼uniformly asymptotic stability 一样渐近稳定性uninterrupted duty 不间断工作制，长期工作制unit circle 单位圆unit testing 单元测试unsupervised learing 非监督学习upper level problem 上级问题urban planning 城市规划utility function 效用函数value engineering 价值工程variable gain 可变增益，可变放大系数variable structure control system 变结构限制vector Lyapunov function 向量李雅普诺夫函数velocity error coefficient 速度误差系数velocity transducer 速度传感器vertical decomposition 纵向分解vibrating wire force transducer 振弦式力传感器vibrometer 振动计viscous damping 粘性阻尼voltage source inverter 电压源型逆变器vortex precession flowmeter 旋进流量计vortex shedding flowmeter 涡街流量计WB (way base) 方法库weighing cell 称重传感器weighting factor 权因子weighting method 加权法Whittaker-Shannon sampling theorem 惠特克-香农采样定理Wiener filtering 维纳滤波work station for computer aided design 计算机协助设计工作站w-plane w平面zero-based budget 零基预算zero-input response 零输入响应zero-state response 零状态响应zero sum game model 零和对策模型z-transform z变换。
SINS初始对准算法综述
SINS初始对准算法综述惯性导航的本质是一个积分航位推算过程，而初始对准的目的就是确定积分初值的过程，包含姿态初值、速度初值和位置初值。
根据对准环境，可将初始对准分为静基座对准、摇摆基座对准和动基座对准。
根据转位方式，又可将静基座对准分为单位置对准、双位置对准和多位置对准，多位置对准可以通过IMU旋转来改善系统的可观测性，从而提高对准和测漂精度。
一、静基座对准载体在静止或微幅晃动条件下，速度初值为零，位置初值通常由外界输入，所以静基座对准的主要任务在于确定载体姿态。
用于姿态描述的数学方法包括方向余弦矩阵（Direction Cosine Matrix, DCM）、欧拉角（Euler Angles）、姿态四元数（Quaternion Symmetric Parameters）、旋转矢量（Rotation Vector）、罗德里格参数（Rodrigues Parameters）或吉布斯向量（Gibbs Vector）、修正罗德里格参数（Modified Rodrigues Parameters, MRPs）、凯莱-克莱参数（Cayley-Klein Parameters）等。
1964年，Stuelpnagel从数学上证明仅采用三维参数表示姿态不可能是全局非奇异的，所以姿态矩阵和四zk元q 数表20示16方02法2使2 用较为普遍，而欧拉角由于物理意义明确也经常使用。
地球表面可供参考的物理矢量为地球自转角速率ie 和重力加速度g ，只要ie 和g 不共线（非极区），就可以通过矢量观测确定最优姿态。
一般认为最优姿态估计算法沿着两条技术路线发展：①确定性算法，即Wahba问题求解[53]；②状态估计法，如卡尔曼滤波方法。
（一）确定性算法1965年Wahba将姿态确定问题描述为带约束的最小二乘估计问题，即Wahba问题。
对于双矢量观测的情况，TRIAD （ThRee-axIs Attitude Determination）方法可以通过直观方式获得闭环解。
微积分教学资料——chapter12
The diffrence u-v of the two vectors u and v. u-v
v
u
Components
The two-dimensional vector aa1,a2 is the position vector of the point P(a1,a2) .
aop P(a1,a2)
x2y2z2r2
z
o
y
x
12.2 Vectors
The term vector is used by scientists to indicate a quantity that has both magnitude and direction.
B Suppose a particle moves along a line
and bthen
a b a b c os0
Proof By the Law of Cosines,we have
a b 2a 2b 22a b co s
Corollary If is the angle between the nonzero
Properties of Vectors
If a,b.and c are vectors in v n and c and d are scalars,
then
1.a+b=b+c
2.a+(b+c)=(a+b)+c
3.a+0=a
4.a+(-a)=0
5.c(a+b)=ca+cb 6.(c+d)a=ca+cd
Distance Formula in Three Dimensions
数学专业英语中英文对照翻译2.5[1]
![数学专业英语中英文对照翻译2.5[1]](https://img.taocdn.com/s3/m/1977e8d37c1cfad6195fa7c1.png)
2.5笛卡尔几何学的基本概念（basic concepts of Cartesian geometry）课文5-A the coordinate system of Cartesian geometryAs mentioned earlier, one of the applications of the integral is the calculation of area. Ordinarily , we do not talk about area by itself ,instead, we talk about the area of something . This means that we have certain objects (polygonal regions, circular regions, parabolic segments etc.) whose areas we wish to measure. If we hope to arrive at a treatment of area that will enable us to deal with many different kinds of objects, we must first find an effective way to describe these objects.The most primitive way of doing this is by drawing figures, as was done by the ancient Greeks. A much better way was suggested by Rene Descartes, who introduced the subject of analytic geometry (also known as Cartesian geometry). Descartes’ idea was to represent geometric points by numbers. The procedure for points in a plane is this ：Two perpendicular reference lines (called coordinate axes) are chosen, one horizonta l (called the “x-axis”), the other vertical (the “y-axis”). Their point of intersection denoted by O, is called the origin. On the x-axis a convenient point is chosen to the right of Oand its distance from O is called the unit distance. Vertical distances along the Y-axis are usually measured with the same unit distance ,although sometimes it is convenient to use a different scale on the y-axis. Now each point in the plane (sometimes called the xy-plane) is assigned a pair of numbers, called its coordinates. These numbers tell us how to locate the points.Figure 2-5-1 illustrates some examples.The point with coordinates (3,2) lies three units to the right of the y-axis and two units above the x-axis.The number 3 is called the x-coordinate of the point,2 its y-coordinate. Points to the left of the y-axis have a negative x-coordinate; those below the x-axis have a negtive y-coordinate. The x-coordinateof a point is sometimes called its abscissa and the y-coordinateis called its ordinate.When we write a pair of numberssuch as (a,b) to represent a point, we agree that the abscissa or x-coordinate,a is written first. For this reason, the pair(a,b) is often referred to as an ordered pair. It is clear that two ordered pairs (a,b) and (c,d) represent the same point if and only if we have a=c and b=d. Points (a,b) with both a and b positiveare said to lie in the first quadrant ,those with a<0 and b>0 are in the second quadrant ; and those with a<0 and b<0 are in the third quadrant ; and those with a>0 and b<0 are in the fourth quadrant. Figure 2-5-1 shows one point in each quadrant.The procedure for points in space is similar. We take three mutually perpendicular lines in space intersecting at a point (the origin) . These lines determine three mutually perpendicular planes ,and each point in space can be completely described by specifying , with appropriate regard for signs ,its distances from these planes. We shall discuee three-dimensional Cartesian geometry in more detail later on ; for the present we confine our attention to plane analytic geometry.课文5-A：笛卡尔几何坐标系正如前面所提到的，积分应用的一种是计算面积。
基于观测向量的四元数姿态估计
X
i
ai bi rT i
(4)
Now kA
so Wahba’s problem is equivalent to the orthogonal Procrustes problem with the proviso that the determinant of A must be +1. The purpose of this paper is to give an overview in a unified notation of algorithms for solving Wahba’s problem, to provide accuracy and speed comparisons, and to present two significant enhancements of existing methods. The popular QUaternion EStimator (QUEST) and EStimators of the Optimal Quaternion (ESOQ and OQ-2) algorithms avoid singularities by employing a rotated reference system. Methods introduced in this paper use information from an a priori quaternion estimate or from the diagonal elements of the B matrix to determine a desirable reference system, avoiding expensive sequential computations. Also, tests show that a first-order expansion in the loss function is adequate, avoiding the need for iterative refinement of the loss function, and motivating the introduction of new first-order versions of ESOQ and ESOQ-2, which are at present the fastest known first-order methods for solving Wahba’s problem. FIRST SOLUTIONS OF WAHBA’S PROBLEM J. L. Farrell and J. C. Stuelpnagel [4], R. H. Wessner [5], J. R. Velman [6], J. E. Brock [7], R. Desjardins, and Wahba presented the first solutions of Wahba’s problem. Farrell and Stuelpnagel noted that any real square matrix, including B , has the polar decomposition B = W R, (7)
threejs 绕指定两点连成的轴旋转
threejs 绕指定两点连成的轴旋转three.js是一款使用JavaScript编写的开源3D图形库，它可以在网页上创建和渲染3D图形。
在three.js中，可以通过旋转物体的角度来实现旋转效果。
要绕指定两点连成的轴旋转物体，我们需要了解three.js中的几何体、材质和旋转相关方法。
首先，让我们来创建一个场景(scene)，在场景中放置一个几何体(geometry)。
在这个例子中，我们选择使用一个立方体(BoxGeometry)。
通过设置立方体的大小、位置和旋转角度，我们可以自定义立方体对象。
```javascriptconst scene = new THREE.Scene();const geometry = new THREE.BoxGeometry(1, 1, 1);const material = new THREE.MeshBasicMaterial({ color:0x00ff00 });const cube = new THREE.Mesh(geometry, material);cube.position.set(0, 0, 0); //设置立方体的位置scene.add(cube);```接下来，我们需要创建一个相机(camera)和一个渲染器(renderer)。
相机用于定义场景在屏幕上的投影位置和视角，而渲染器则用于将场景渲染到屏幕上。
```javascriptconst camera = new THREE.PerspectiveCamera(75,window.innerWidth / window.innerHeight, 0.1, 1000);const renderer = new THREE.WebGLRenderer({ antialias: true });renderer.setSize(window.innerWidth, window.innerHeight);document.body.appendChild(renderer.domElement);camera.position.z = 5;function animate() {requestAnimationFrame(animate);//旋转代码renderer.render(scene, camera);}animate();```现在，让我们来实现绕指定两点连成的轴旋转立方体的效果。
极角 方位角 英语
极角方位角英语Polar Angle and Azimuth in EnglishThe concept of polar angle and azimuth are fundamental to understanding the spatial positioning of an object in athree-dimensional coordinate system.The polar angle,also known as the zenith angle or inclination angle,represents the angle between the positive z-axis and the vector that points to the object.It is measured in degrees or radians and ranges from0°to180°.On the other hand,the azimuth angle,also known as the bearing or horizontal angle,represents the angle between the positive x-axis and the projection of the vector onto the xy-plane.It is measured in the xy-plane from the positive x-axis and ranges from0°to360°.In practical applications,understanding the polar angle and azimuth of an object is crucial for various fields such as astronomy,navigation,and robotics.By knowing these angles,one can determine the exact orientation of an object relative to a reference point or coordinate system.In conclusion,the concepts of polar angle and azimuth play a key role in defining the spatial position of an object in a three-dimensional space.By understanding and applying these concepts,one can accurately describe and analyze the orientation of objects in a variety of fields.。
Three.js中矩阵和向量的使用教程
Three.js中矩阵和向量的使⽤教程前⾔提起矩阵，很容易让⼈想起我们曾经学不会的线性代数和离散数学，但是作为图形开发中的核⼼部分，它代表着每⼀次的运动和变换，就像鱼不能脱离⽔⼀样，矩阵并不是⼀个可以避之不谈的话题。
好消息是，Three.js帮助我们把许多矩阵运算封装成了⼀些顶层的⽅法，并提供了⼀个优秀的数学库，我们不太需要知道HowToCalc，只需要知道HowToUse，就可以得到绝⼤部分我们想要的东西。
这篇⽂章将要介绍的就是，如何在不了解内部结构的情况下在Three.js中使⽤矩阵和向量。
从⼀个例⼦开始在讲解⼀些枯燥的概念前先举⼀个⼩例⼦，来简要说明⼀下为什么我们要使⽤矩阵⽅法。
这是我们最终要完成的效果。
⾸先，我们要创建三个⼏何体：var box_geometry = new THREE.BoxGeometry();var sphere_geometry = new THREE.SphereGeometry(0.5, 32, 32);var cylinder_geometry = new THREE.CylinderGeometry(0.1, 0.1, 0.5);var material = new THREE.MeshLambertMaterial({color: new THREE.Color(0.9, 0.55, 0.4)})这三个⼏何体分别是盒⼦、球和圆柱体。
然后去创建三个⽹格，并将它们置⼊场景。
var box = new THREE.Mesh(box_geometry, material);var sphere = new THREE.Mesh(sphere_geometry, material);sphere.position.y += 1;var cylinder = new THREE.Mesh(cylinder_geometry, material);cylinder.position.y += 1.75;scene.add(box);scene.add(sphere);scene.add(cylinder);这段代码将⽣成如下场景:虽然不那么美观，但作为⽰例已经⾜够了，现在我希望这堆物体尺⼨减半。
threejs位置和旋转的详细解析-概述说明以及解释
threejs位置和旋转的详细解析-概述说明以及解释1.引言【1.1 概述】概述部分将对整篇长文进行简要介绍，包括文章涉及的主题、研究的对象以及解析的目标。
本文的主要目标是对Three.js中的位置和旋转进行详细解析，帮助读者更好地理解和应用这两个重要的概念。
首先，位置是指物体在三维空间中的坐标点，用于描述物体在空间中的位置关系。
在Three.js中，位置通常以三维向量的形式表示。
了解位置的概念和相关的计算方法，对于实现物体的移动和定位非常重要。
其次，旋转是指物体在三维空间中的方向变化，可以视为物体绕着一个中心点进行转动。
在Three.js中，旋转通常以四元数、欧拉角或旋转矩阵的形式表示。
理解旋转的概念和不同的表示方法，可以帮助我们控制物体的朝向和角度。
本文将分为多个章节，逐步介绍位置和旋转的概念、表示方法以及计算方法。
在正文部分，首先对位置进行详细解析，包括位置的定义、表示方法以及计算方法。
然后对旋转进行详细解析，包括旋转的定义、表示方法以及计算方法。
最后，探讨了位置和旋转的关系，包括复合变换、应用场景以及局限性。
总之，掌握位置和旋转的原理和技术，对于进行三维空间的建模和动画制作非常重要。
本文旨在为读者提供一个全面且详细的位置和旋转解析，希望读者通过本文的阅读和学习能够更好地理解和应用Three.js中的位置和旋转。
1.2文章结构1.2 文章结构在本篇长文中，我们将详细解析three.js中的位置和旋转，并探讨它们之间的关系。
文章主要分为三个部分：引言、正文和结论。
引言部分将为读者提供一个概述，介绍了本文要讨论的主题以及文章的目的。
我们将简要介绍three.js是什么，以及为什么位置和旋转在three.js中如此重要。
通过引入这些基本概念，读者将对以下内容有更好的理解。
正文部分将是本文的核心内容，包含了位置和旋转的详细解析。
我们将从位置开始，首先解释什么是位置，它在three.js中的作用以及为什么它是三维场景中不可或缺的。
Bartington Instruments Mag-13三轴磁场传感器操作手册说明书
Operation Manual for Mag-13 Three-Axis Magnetic Field Sensors1. About this Manual 31.1. Symbols Glossary 32 Safe Use 33. Introduction 34. General Description 45. Enclosures 46. Compatible Power Supply and Data Acquisition Units 47. Cables and Connectors 57.1. Cables 57.2. Mating Connectors 58. Mounting 58.1. Mag-13MC 68.2. Mag-13MS 69. Operation 79.1. Connector Pin Allocation 79.2. Interface 79.3. Power Supplies 79.4. Signal/Power Ground 79.5. Test Coil 89.6. Temperature Sensor 99.7. Connecting Power 99.8. Electromagnetic Compatibility 910. Performance 910.1. Noise 910.2 Excitation Frequency 1011. Troubleshooting, Care and Maintenance 1011.1. Troubleshooting 1011.2. Care and Maintenance 1012. Storage & Transport 1013. Disposal 1113.1. Waste Electrical and Electronic Equipment (WEEE) Regulations 111.This manual describes the installation, operation and maintenance of the Mag-13 range of three axis magnetic field sensors. It should be read in conjunction with the product brochure DS3143 and the outline drawings which can be found on the Mag-13 product page on the Bartington Instruments website at: .See Application Note AN0045: ‘Magnetic Units and Measurements’, available from Bartington Instruments, for important information about magnetic field measurement units.The following symbols used within this manual call your attention to specific types of information: WARNING: Indicates a situation in which serious bodily injury or death could result if thewarning is ignored.Caution: Indicates a situation in which bodily injury or damage to your instrument, or both,could result if the caution is ignored.Identifies items that must be disposed of safely to prevent unnecessary damage to theenvironment.Note: Provides useful supporting information on how to make better use of your purchase. 2WARNING: These products are not qualified for use in explosive atmospheres or life supportsystems. Consult Bartington Instruments for advice.These compact, high performance sensors with integral electronics provide measurementsof static and alternating magnetic fields in three axes. The sensors, also described as magnetometers, convert magnetic flux density, measured in three axes, into a bipolar analogue voltage. Analogue output voltages Vx, Vy and Vz vary linearly with magnetic flux density.The Mag-13 series are an evolution of the Mag-03 range of sensors and feature a number of improvements. These include improved orthogonality error of the sensing coils, and an inbuilt test coil and temperature sensor. There have also been other refinements in the design.The analogue output is positive for conventional flux direction, South to North, in the direction of the arrow shown on the label for each axis; i.e. the maximum positive output will be obtained from any axiswhen the arrow points towards magnetic north along the total field vector. The measurement axes are designated X, Y and Z in the Cartesian co-ordinate system, when viewed from the top or non-connector end of the sensor.The analogue outputs may require external filters if not used with a Bartington Instruments data acquisition unit, to achieve the noise specification of the sensor (see Noise).4.The Mag-13 contains three fluxgate sensing elements mounted orthogonally in a block at one end of an enclosure, which also contains the electronic circuitry. The connector is mounted at the opposite end of the enclosure. The position and direction of each sensing element is indicated by arrows on the outside of the sensor, together with the product code, measuring range and serial number. The sensor elements are precisely aligned along the centre lines of the package, directly beneath the diagram on the label.Details of the enclosures, mounting, connector dimensions, connector pin allocation and the position of the sensing elements relative to the enclosure are given in outline drawings on the Mag-13 product page.The sensors provide three high precision analogue outputs, proportional to the magnetic field along each axis. The relationship between the magnetic field and the analogue output is extremely linear. An additional output is provided in order to take temperature measurements or activate the internal test coil.The low output impedance of the sensor ensures it can be operated over long cables when interfaced with Bartington Instruments’ high impedance data acquisition systems. The zero field offset error, scale factor, orthogonality and frequency response are individually calibrated.The sensors are available in a variety of enclosures. All enclosures are environmentally sealed and electrically shielded. A full list of sensors with specifications is provided in the product brochure.Note: Using your sensor in an environment that exceeds its rating may result in the need for repair at the customer’s expense.6.A number of other Bartington Instruments products will work with the Mag-13 as power supply and/or data acquisition units. These are listed in the product brochure and can be found at www. /data-acquisition-and-conditioning-units.html.Note: Outputs for the test coil and temperature sensor are presently only available with the DecaPSU Power Supply Unit. See Test Coil for information on test coil operation.For further information on power supplies see Power Supplies.For information on using your own power supply or data acquisition unit see AN0042: ‘Connecting your own Power Supply to a Bartington Magnetic Field Sensor’, available from Bartington Instruments.Although the Mag-13 is singled-ended, the cable connecting it to the power supply has a signal ground for each axis. The configuration of the cable pin out for the Mag-13 on the power supply side will require the power supply/signal conditioning unit to be set to the balanced mode to operate the Mag-13 correctly.For the Spectramag and Mag-03DAM data acquisition units, an adaptor cable will be required. This cable must be used despite the connector of the original cable being compatible with the power supplies.Caution: The Mag-13 sensor will not function correctly if plugged directly into aSpectramag-6 or Mag-03DAM without the use of an adaptor cable.7.Cables are available to connect the range of Mag-13 sensors to the range of suitable Bartington Instruments power supply and data acquisition units. Specifications for each of the cables are given in the product brochure.Note: Cables must be ordered separately.Note: Customers manufacturing their own cables must ensure the cables are shielded to prevent them picking up EM (electromagnetic) interference.7.2.For information on suitable mating connectors refer to the product datasheet.The range of compatible mounting accessories are shown in the product brochure.The method of mounting will depend on the application and the enclosure. For details of the mounting arrangements for each sensor, refer to the product brochure and the relevant outline drawing on the Mag-13 product page.Note: The use of magnetic materials in the mounting arrangement must be avoided. Check all mounting components before installation by placing the component within the immediate vicinity of the sensing elements of a working magnetometer and observing any variation in the background field.Caution: Do not place the sensor head of the unpackaged sensor in the immediate vicinity ofelectrically conductive materials.Caution: The absolute maximum screw penetration depth within the body, as shown in therelevant outline drawing on the product page, must not be exceeded.The analogue output is positive for conventional flux direction, south to north, in the direction of the arrow shown on the label for each axis; i.e. the maximum positive output will be obtained from any axis when the arrow points towards magnetic north along the total field vector.8.1.This sensor can be supported by the Mag-TA Tripod Adaptor described in the product brochure.The end of the Mag-13MC has a threaded hole for fixing the sensor in place, and three conical indentations that can be used to achieve a more precise alignment of the axes when setting up the sensor.Caution: The label area of the sensor is recessed and should not be used for clamping.8.2.This sensor can be supported by the Mag-TA Tripod Adaptor described in the product brochure. This sensor has threaded holes tapped in the base which is also the datum face. The sensor can be mounted on any flat, non-magnetic surface, including the Tripod Adaptor, using the two brass screws supplied.Caution: The absolute maximum screw penetration depth within the body is 8 mm and thismust not be exceeded.9.The connector pin or cable colour allocation for the connection to each package type is shown on the appropriate outline drawing on the Mag-13 product page.The analogue outputs for the X, Y and Z axes are buffered to give a low output impedance, enabling the unit to be operated over long cables and interfaced with high impedance data acquisition systems.The normal power supply of the sensors is specified in the product brochure. The ideal power supply units are those referenced in Compatible Power Supply and Data Acquisition Units. Alternatively, users may wish to provide their own supply. This should provide a voltage within the specification found in the product brochure. For the low noise applications, any ripple in the power supply should not exceed a few mV.Note: Adequate performance of the sensor cannot be guaranteed if used with non-Bartington Instruments products. Bartington Instruments cannot advise on the operation of third party products.See the product brochure for nominal current requirements. There is an additional current in proportion to the measured field, which is drawn from the positive or negative supply depending on the direction of the field.Note: The two signal/power ground conductors are connected to a common point within the sensor and the power supply common (power 0V) should be connected to only one of them.The other signal/power ground conductor should be used as the signal output common (0V).Each signal is then measured between the signal output conductor and the signal output common. In this way, the signal output common carries no power supply currents.Note: In long cables, the minimum current in the power ground conductor will give rise to an appreciable potential difference between the power supply end and the sensor end of the power ground conductor. The use of separate power and signal ground conductors will ensure thatthis voltage is not included in the voltage measured between the signal output and the signal common.Wiring for the Mag-13 is shown in the diagram below.Mag-13 outline wiring diagramTest CoilThe Mag-13 features a test coil which applies a magnetic field to all axes when activated. The corresponding changes in X, Y and Z output voltages confirm that all axes of the sensor are operational.The test coil can be activated by grounding the relevant pin of the Mag-13 output connector, i.e. connecting it to 0V. In this way it can be seen that the sensor is measuring correctly when the corresponding change in field is detected in each of the three axes. Refer to the test record for the field values generated for each individual sensor.Note: Of the available Bartington power supplies, only the Decaport and the DecaPSU will allow operation of the test coil.The Mag-13 features an inbuilt temperature sensor that can be used to measure temperature changes inside the sensor over time while taking measurements.The temperature can be read from the same pin as the test coil as an analogue voltage output. Measure the output between the relevant pin and signal ground. Refer to the product brochure for the voltage output and temperature scaling parameters.Caution: Check that the polarity of the supply is correct. Incorrect polarity can be preventedby using the power supply and cables provided by Bartington Instruments.Caution: The power supply should be connected to the sensor before the supply isenergised, as this prevents high surge currents which could cause damage. Apply thepositive and negative supplies simultaneously and avoid leaving the sensor connected to one polarity only.9.8.The Mag-13 range of sensors are electrically shielded from external, and emission of internal, electromagnetic fields. Any emissions generated are at a low level with a primary frequency corresponding to the frequency of the energising field of the sensor. The sensor is required to respond to magnetic fields within the specified frequency band.Caution: Do not operate the sensor in very strong electromagnetic fields as it may develop apermanent offset, or damage could occur to the sensing coils.Note: Do not place the sensor near to any equipment which may be affected by the verysmall local field produced by the sensor excitation.10. PFor detailed figures on the performance of the Mag-13 range of sensors, refer to the product brochure.N oiseThe Mag-13 range includes different noise versions that are specified in the product brochure. These versions correspond to the internal noise of the sensor, which can only be achieved in a shielded environment where no external fields are present.10.2The output signal for each axis will also contain breakthrough, which is a residual signal associated with the excitation frequency. (See the product brochure for frequency and level of breakthrough.) All Bartington Instruments power supply and signal conditioning units have a filter to remove the breakthrough.Note: When using a non-Bartington Instruments power supply, it will be necessary to providea filter to remove the breakthrough. Not doing so will lead to a higher noise level than thatspecified. See application note ‘AN0042: Connecting your own Power Supply to a BartingtonMagnetic Field Sensor’ from Bartington Instruments for further information.11.Special equipment is required for the diagnosis of faults within the unit. Much of this equipment is beyond the scope of normal service facilities. Therefore, in the event of any apparent malfunction, email service@ or telephone the Bartington Instruments service team on +44 (0)1993 706565.Caution: Attempted repair or opening of the casing by users may invalidate the warranty.A calibration service is available from Bartington Instruments which is traceable to international standards.Surface or dirt contamination should be removed using a mild detergent solution only. If the connector pins become contaminated then they should be lightly cleaned with a swab of isopropyl alcohol.Note: Dirt on the connectors may lead to increased noise in the output.12. SYour sensor is a precision electronic instrument and should be treated as such.Caution: Avoid exposing this intrument to shocks or continuous vibration.Caution: Store only within the temperature range specified in the product brochure.Caution: Do not expose this instrument to strong magnetic fields while being stored.BARTINGTON INSTRUMENTSPage 11 of 12 OM3143/313.This product should not be disposed of in domestic or municipal waste. For information about disposing of your sensor safely, check local regulations for disposal of electrical / electronic products.Bartington Instruments Mag-13 sensors comply fully with Restriction of the Use ofCertain Hazardous Substances in Electrical and Electronic Equipment (RoHS) and WEEERegulations current at the time of printing.O M 3143/3The copyright of this document is the property of Bartington Instruments Ltd.Bartington® is a registered trade mark of Bartington Instruments Limited in the following countries: United Kingdom, Australia, Brazil, Canada, China, European Union, India, Japan, Norway and the United States of America.Bartington Instruments Limited 5 Thorney Leys Business Park,Witney, Oxford, OX28 4GE, England. T: +44 (0)1993 706565F: +44 (0)1993 774813E: sales @。
多矢量定姿的SVD和QUEST算法等价性分析
多矢量定姿的SVD和QUEST算法等价性分析严恭敏; 陈若彤; 郭鹍【期刊名称】《《中国惯性技术学报》》【年(卷),期】2019(027)005【总页数】5页(P568-572)【关键词】捷联惯导系统; 姿态确定; 奇异值分解; QUEST估计方法【作者】严恭敏; 陈若彤; 郭鹍【作者单位】西北工业大学自动化学院西安 710072; 中船航海科技有限责任公司北京 100070【正文语种】中文【中图分类】V249.3捷联惯导系统初始对准的重点在于初始姿态阵的求解，它通常可分为滤波估计和确定性算法两种方法[1]。
滤波方法通过建立系统误差模型，模型中往往包含了惯性传感器误差，在系统建模准确的情况下一般具有较好的对准精度。
然而，滤波方法通常只有在小失准角误差线性建模条件下才会比较有效，虽然大失准模型及其非线性滤波方法研究较多，但实际工程使用的却比较少。
采用大失准角模型通常也只是为了获得粗略的失准角，再转而使用小失准角线性模型，才能得到高精度的初始对准结果[2]。
确定性姿态算法与模型是否线性无关，在航天器姿态确定中应用较多，它同样也适用于捷联惯导的粗对准过程，有时在惯导机动干扰不大的场合也能获得较高的初始对准精度，甚至达到精对准的效果[3-4]。
姿态确定算法早期研究主要源于解决卫星姿态的确定问题。
1964年Harold Black提出了利用两组观测矢量求取姿态转换矩阵的代数方法TRIAD（Tri-Axial Attitude Determination，常称为双矢量定姿法）；1965年Grace Wahba提出了应用观测矢量进行星体姿态确定的最小二乘优化问题（常称为 Wahba问题）[5]；1968年Davenport提出了q-method 方法，采用四元数姿态描述方式解决 Wahba问题，或称 QUEST估计方法（QUaternion ESTimator）[6]。
由于早期计算机的运算能力有限，研究者们提出了一系列改进的四元数优化快速算法，比如FOAM（Fast Optimal Attitude Matrix）、ESOQ（EStimator of the Optimal Quaternion）、ESOQ2（Second ESOQ）等算法，主要用于降低计算量和提高数值鲁棒性[7-10]。
GUIDANCE AND CONTROL OF ARTILLERY PROJECTILES WITH MAGNETIC SENSORS
23RD INTERNATIONAL SYMPOSIUM ON BALLISTICSTARRAGONA, SPAIN 16-20 APRIL 2007GUIDANCE AND CONTROL OF ARTILLERY PROJECTILESWITH MAGNETIC SENSORSPhilippe RougerNexter Munitions, Ballistics and Flight Control Department , 7 route de Guerry, 18000 Bourges CedexThis article deals with a new system to ensure Guidance and Control (G&C) of an artillery projectile without the use of a complete standard Inertial Measurement Unit system (IMU). Nexter Munitions is interested in new technology based on magnetoresistive sensors due to their size, high resistance to extremely severe environments and their price. This new system replaces the three axis gyros of the inertial measurement unit by a three axis magnetic sensor unit (“gyrofree IMU”) strapped down in the principal axis of the projectile. It is called an “electronic compas”The system is based on the hypothesis that the “terrestrial magnetic field” vector, at a specific date, remains constant for several miles around the firing position. In flight, the real time comparison of this field and the magnetic field measured in projectile axis provides the attitude and rotation rate of the projectile.A real time algorithm has been developed to estimate and correct magnetic disturbances in flight.The performance of such a system is improved by the use of the accelerometers. First estimations show that miss distance precision could then be to less than a few meters in approximately more than 98% of the cases : as long as the longitudinal axis of the projectile is out side of a 10° cone around the local earth magnetic field vector. This level could be increased if the scenario is adjusted to geographic localisation of firing.In parallel with guidance and control algorithms development, Nexter Munitions has been conducting demonstrations.NOMENCLATUREX O, Y O, Z O space -fixed system of axis, positioned at firing pointX P, Y P, Z P projectile body-fixed system of axis, at the projectile center of gravityH local earth magnetic field (3-D vector)EH earth magnetic field (3-D vector), in space-fixed system of axisH earth magnetic field (3-D vector), in body-fixed system of axisPH O_X, H O_Y , H O_Z X, Y and Z components of 0HH P_X, H P_Y, H P_Z X, Y and Z components of P H789EXTERIOR BALLISTICS790H P_T transversal magnetic field in body-fixed system of axisB Hp_x bias estimation for longitudinal body-fixed axisψ, θ, ϕEuler’s angles of yaw, pitch and roll.M OP transformation matrix from space to body-fixed axis θ&ϕ&yaw, pitch and roll rate of change in fixed system of axisψ&, ,p, q, r roll, pitch and yaw rotation rates in body-fixed axisfunctionsb, c, d, e, f, g genericεangle of the cone around axis of Earth magnetic field vector EINTRODUCTIONThis article deals with a new system to ensure guidance and control of an artillery projectile without the use of a classical IMU system including accelerometers and gyros. In spite of recent technical progress in inertial measurement unit systems, it is still difficult to develop accurate gyros supporting artillery accelerations which can reach more than twenty thousand ‘g’. In addition, the few systems that actually appear in this field are promising but still remain very expensive.So the objective is to develop a system which can ensure the same functions as the gyros for guidance and control. This means that this system has to provide the autopilot with data related to projectile attitude and its angular rates during the different phases of the flight.With these technical and operational constraints, Nexter Munitions is interested in the new technology based on magnetoresistive sensors : these are miniaturized, g-hardened2, cheap and off-the-shelf. This objective has been reached and the three axis gyros of the inertial measurement unit can be replaced by a three axis magnetic strapped down sensor.This article presents the hypothesis and the method developed. Then, it describes the flight scenario defined to evaluate the availability to guide and control a projectile with the electronic compass. First performance estimations is presented, followed with limitation of use, mainly depending on geographical location on Earth. The last part presents the first steps of experimentation of this system.DEVELOPMENT OF THE METHODThe basic hypothesis of this concept is to consider that the “earth magnetic field” vector E, at a given time, remains constant within a radius of several miles around the firing position1. This information can be stored before firing in a specific memory of the projectile data-processor (called0) and compared at each time with the measurements (called P H) implemented during the flight :Guidance and control of artillery projectiles with magnetic sensors 7910point H P =magnetic field measurement in flight, in projectile principal axesHypothesis for this relationship :-a few miles around launching point-a few minutes after H 0measurement H PH 0Figure 1. Electronic compass basic hypothesis : Earth magnetic field E is constant in flight. The relationship used for the comparison of 0 and P is classical :P OP H M H ∗=0 (1)This relationship reveals the M OP rotation matrix which only depends on the projectile attitude angles. As Whaba 3 demonstrated it in the seventies, the eq. (1) system has an infinite number of solutions. It needs additional information to be reduced to a unique solution.Without additional sensor, it is necessary to find something else to solve this problem. For a typical artillery projectile flight, we can observe that the yaw angle remains constant during the ballistic phase of the flight. In fact, during such a so short flight (less than a few kilometers), the influence of the Coriolis force can be neglected. Based on this a priori knowledge of the yaw angle, the eq. (1) can be solved during ballistic flight and also during a pre-programmed controlled flight. The dynamic behaviour of the projectile is supposed to be known with the required level of accuracy. Thus, yaw angle evolution is estimated before flight and stored in the onboard calculator memory. Pitch angle can be obtained with H P_X or, but as for yaw, it could be obtained with pre flight simulation. This solution will be preferred and pitch evolution will be stored in the onboard calculator memory. Roll angle “ϕ” is obtained with transversal measurements H P_Y and H P_Z :⎜⎝⎛⎟⎠⎞+=.e .d .d - .e atan P_Y P_Z P_Z P_Y H H H H ϕ (2)with (d, e) functions of yaw angle, pitch angle and 0.This straightforward step by step resolution gives all attitude angles of the projectile in real time, without requiring estimation techniques which have typically a long time of convergence. Last step to obtain angular rate of the projectile is to solve the system below :EXTERIOR BALLISTICS792 ).sin( - )cos( / ) )q.sin( )cos(. ( )q.cos( - )sin(.θψθϕϕϕϕϕψθ&&&&p r r += (3)Before such a resolution, it is necessary to verify that all the expression rates are valid. Roll angle estimation is restricted by arctangent limitations to +/-2π variations. This induced a sudden discontinuity for roll estimations near 2π values. A correction is done for each detection of an extreme difference (roughly equals to 2π) between two successive roll angle estimations. After several attitude estimations, rotation rates are given to the autopilot to ensure navigation, guidance and control without the assistance of gyros.Projectile’s attitude estimation can be greatly improved by the use of the three axis accelerometers. Yaw and pitch angle estimations can be obtained by the use of acceleration measurements carried out during the flight 7. You can see below a comparison of ψ and θ angles obtained with the two methods and with real attitude angles during the flight :Figure 2. In-flight true yaw and pitch angles compared with both algorithm estimationsAll these estimations are directly delivered to the autopilot, thus navigation with an “electronic compass” is slightly different from navigation with a standard IMU with gyros. Rotation rate estimates are only used to stabilise the projectile yaw, pitch and roll channel control loop.Guidance and control of artillery projectiles with magnetic sensors 793 PERFORMANCE EVALUATIONScenarioEvaluation of the electronic compass capability for guidance and control will beFigure 3. Typical artillery controlled flightDuring the ballistic flight, the trajectory in the vertical plane is curved under the effect of gravity and the horizontal trajectory is a straight line. A lateral aiming error is introduced in order to require a correction of the trajectory to the right during the flight to reach the target. At the same time, flight altitude is increased to reach a better vertical angle for the terminal attack.Yaw and pitch estimations could be obtained with a pre-flight calculation, and stored onboard as a time law evolution, or with accelerometers measurements. Implementation of Magnetic MeasurementsThis system can be very accurate if good measurements are available. So the magnetic sensors have to be protected as high as possible form disturbances. But the magnetic environment of the projectile is neither clean nor well known and sources of disturbance are numerous : permanent, induced magnetism and electrical current disturbance. Due to the very restricted volume available in the shell, magnetic sensors can’t be placed in magnetically non disturbed area. Misalignment, resulting from sensor cross axis6 and assembling precision reduce accuracy of attitude estimations.The hereby proposed solution is a light ground calibration. Many techniques are available and presented in public sources4,5. All the results of this calibration will be stored onboard and in-flight measurements will be corrected by software. This should be sufficient, but the effect of firing on sensor sensitivity is quite unpredictable. Thus anEXTERIOR BALLISTICS794in flight calibration should be done. This is based on 0 estimation, by software modeling 2 or by measurement before firing. H P_X and H P_T can be calculated before firing and their evolution during the flight is expressed as a simple expression of time :)(),,(²²0___t g H f H H H Z P Y P T P =+=θψ (4)For the correction of the transverse measurements, the shape of the projectile is designed to ensure a required value of roll rate in flight. As the projectile rolls around its longitudinal axis, transverse magnetic measurements H P_Y and H P_Z are sinus. Comparison of the measurements and expected values H P_T of the peak to peak sinus signal gives us the bias and scale factor of transversal magnetic measurements 4.Thus, the complete magnetic measurement correction will be done with pre-flight ground calibration plus in-flight calibration.Performance and Restriction Of UsePerformance assessment of this concept has been carried out, taking into account many sources of disturbances : meteorology, ballistic errors and, of course, magnetic. These evaluations have been done for different firing locations, such as near the Equator or in Central Europe, for different target azimuth angles.First evaluations have been carried out for targets at short ranges, around three and a half kilometers from firing position, with a strap down seeker delivering measurements at a sampling rate of 20 Hz. These result show that the Circular Error Probability (CEP) of the miss distance could be reduced to less than a few meters (fig.4) if the true transverse magnetic signal available is significant compared withdisturbances :Figure 4. CEP for short distance targetACCURACYMAXIMUMAREA Figure 5. Restriction of useThe main restriction is encountered when the projectile flight direction is roughly the same as the earth magnetic field vector E H . For such occasions, roll estimation is most sensitive to small magnetic disturbance. This defines approximately a 10 degrees “blind cone” around local E (fig. 5) : it represents less than 2% of the global firing situations.Guidance and control of artillery projectiles with magnetic sensors 795A possible way to improve electronic compass performance is to adapt the operational flight solution depending on the local earth magnetic field orientation. This can be done because of the maneuver capabilities of the projectile.EXPERIMENTATIONIn parallel with the guidance and control algorithms development, Nexter Munitions has been conducting demonstrations from hardware in the loop test to firing experimentation (fig.6). Firing tests have been carried out with a 120 mm tank gun and initial acceleration of more than 15000 “g”. The three magnetic measurements have been recorded during ballistic flight and analyzed after recovery. Pre flight calibration of magnetic sensors was simple with a rough estimation of scale factor and bias after integration into the shell. A video tracking system (fig.7) was involved to estimate the projectile roll rate in flight :Figure 6. electronic compass integrated in a shell Figure 7. Extract of the video tracking The measurements have been recovered and a subsequent roll rate estimation have been deduced from signal analysis. Longitudinal axis measurement fluctuations (blue line on fig. 8) are typical of non-orthogonality and misalignment. The fitting of the integration of sensors has to be improved. Transverse Y and Z oscillation around non null value reveals bias. All these distortions directly induce roll rate fluctuations compared to the roll rate estimate based on the video tracking. The difference between the two roll rate estimations (fig. 9) is only about a third revolution per second (10% max around the nominal value obtained from the video analysis). This means that the theoretical studies and developments are promising7 to allow the flight control of an artillery shell.EXTERIOR BALLISTICS 796Figure 8. Three magnetic sensors measurements Figure 9. Roll rate estimations CONCLUSIONSThe objective of Nexter munitions was to propose a guidance and control concept meeting the requirements of a guided artillery shell in terms of performance, robustness and cost without the use of gyros ("gyrofree IMU"). This solution is based on three magnetic sensors and complies with the aforesaid objectives. Onboard magnetic measurements are compared in real time with the local magnetic field observed at the firing point. Then the attitude of the projectile (Euler's angles) and its rotation rate (p, q, r) are obtained thanks to the use of robust algorithms. The accuracy is significantly improved with the assistance of the three axis accelerometers. The required accuracy can be reached in more than ninety two percent of firing scenarii. The "blind cone" effect can be minimised if operational flight solutions is adapted to local terrestrial magnetic field orientation. The studies described in this article are still continuing in Nexter Munitions.REFERENCES[1] IGRF, The International Geomagnetic Reference Field, British Geophysical survey. Natural Environment Research Council, URL: /models/cgm/cgm.html .[2] Kreitz J, Junod E, Schmoltzi D, Sommer E, Fleck V, Gastebois G.‚ “Sensors and Telemetry System for On-Board Motion Measurements in a 120 mm Kinetic Energy Projectile,” French-German Research Institute of Saint Louis. Year 2004.[3] Grace Wahba, “At least square Estimate of satellite Attitude (65-1 solution),” SIAM Review, vol 8, 1966.[4] Caruso Michael J., “Applications of Magnetic Sensors for Low Cost Compass Systems,” Honeywell, SSEC.[5] Gouws DJ Van Der Merwe EF, “The Implementation of a Magnetic Sensor On Dynamic Platform,” Hermanus Magnetic Observatory.[6] Bharat B Pant, Caruso M., “Magnetic sensor Cross-Axis Effect, “ AN-205, Honeywell publication.[7] Ph. Rouger, Nexter Munitions, FR, Patent Application for a “Procédé permettant d'assurer la navigation et/ou le guidage et/ou le pilotage d'un projectile vers un objectif,” Application No. FR05-13205, filed 19 Dec. 2005.。
使用threejs实现第一人称视角的移动的问题（示例代码）
使⽤threejs实现第⼀⼈称视⾓的移动的问题（⽰例代码）在数据可视化领域利⽤webgl来创建三维场景或VR已经越来越普遍，各种开发框架也应运⽽⽣。
今天我们就通过最基本的threejs来完成第⼀⼈称视⾓的场景巡检功能。
如果你是⼀位threejs的初学者或正打算⼊门，我强烈推荐你仔细阅读本⽂并在我的代码基础之上继续深⼊学习。
因为它将是你能够在⽹上找到的最好的免费中⽂教程，通过本⽂你可以学习到⼀些基本的三维理论，threejs的api接⼝以及你应该掌握的数学知识。
当然要想完全掌握threejs可能还有很长的路需要⾛，但⾄少今天我将带你⼊门并传授⼀些独特的学习技巧。
第⼀⼈称视⾓的场景巡检主要需要解决两个问题，⼈物在场景中的移动和碰撞检测。
移动与碰撞功能是所有三维场景⾸先需要解决的基本问题。
为了⽅便理解，⾸先需要构建⼀个简单的三维场景并在遇到问题的时候向你演⽰如何解决它。
<!DOCTYPE html><html lang="en"><head><meta charset="UTF-8"><title>平移与碰撞</title><script src="js/three.js"></script><script src="js/jquery3.4.1.js"></script></head><body><canvas id="mainCanvas"></canvas></body><script>let scene, camera, renderer, leftPress, cube;init();helper();createBoxer();animate();function init() {// 初始化场景scene = new THREE.Scene();scene.background = new THREE.Color(0xffffff);// 创建渲染器renderer = new THREE.WebGLRenderer({canvas: document.getElementById("mainCanvas"),antialias: true, // 抗锯齿alpha: true});renderer.setSize(window.innerWidth, window.innerHeight);// 创建透视相机camera = new THREE.PerspectiveCamera(75, window.innerWidth / window.innerHeight, 0.1, 1000);camera.position.set(0, 40, 30);camera.lookAt(0, 0, 0);// 参数初始化mouse = new THREE.Vector2();raycaster = new THREE.Raycaster();// 环境光var ambientLight = new THREE.AmbientLight(0x606060);scene.add(ambientLight);// 平⾏光var directionalLight = new THREE.DirectionalLight(0xBCD2EE);directionalLight.position.set(1, 0.75, 0.5).normalize();scene.add(directionalLight);}function helper() {var grid = new THREE.GridHelper(100, 20, 0xFF0000, 0x000000);grid.material.opacity = 0.1;grid.material.transparent = true;scene.add(grid);var axesHelper = new THREE.AxesHelper(30);scene.add(axesHelper);function animate() {requestAnimationFrame(animate);renderer.render(scene, camera);function createBoxer() {var geometry = new THREE.BoxGeometry(5, 5, 5);var material = new THREE.MeshPhongMaterial({color: 0x00ff00});cube = new THREE.Mesh(geometry, material);scene.add(cube);$(window).mousemove(function (event) {event.preventDefault();if (leftPress) {cube.rotateOnAxis(new THREE.Vector3(0, 1, 0),event.originalEvent.movementX / 500);new THREE.Vector3(1, 0, 0),event.originalEvent.movementY / 500}});$(window).mousedown(function (event) {leftPress = true;$(window).mouseup(function (event) {leftPress = false;</script></html>很多js的开发⼈员⾮常熟悉jquery，我引⽤它确实让代码显得更加简单。
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I.
Introduction
RECURRENT problem in spacecraft attitude determination is to determine the attitude from a set of vector measurements. Thus, an orthogonal matrix A (the attitude matrix or direction-cosine matrix) is sought which satisfies
70
AIAA 81-4003
J. GUIDANCE AND CONTROL
VOL. 4, NO. 1
Three-Axis Attitude Determination from Vector Observations
M. D. Shuster* and S.D. Oht Computer Sciences Corporation, Silver Spring, Md.
A
A V; =
(D
where F;,..., Vn are a set of reference unit vectors, which are n known directions (e.g., the direction of the Earth, the sun, a star, or the geomagnetic field) in the reference coordinate system, and W1)...,Wn are the observation unit vectors, which are the same n directions as measured in the spacecraftbody coordinate system. (In general, unit vectors will be denoted by carets.) Because both the observation and the reference unit vectors are corrupted by error, a solution for A does not exist in general, not even for n = 2. This work studies two approaches to this problem: one deterministic and the other optimal. The deterministic method, the TRIAD algorithm, 1 determines the attitude by first discarding part of the measurements so that a solution exists. Because the algorithm is very simple, it has become the most popular method for determining three-axis attitude for spacecraft that provide complete vector information. The algorithm has been in existence for at least a decade and has been implemented in many recent missions. These have included Small Astronomy Satellite (SAS), Seasat, Atmospheric Explorer Missions (AEM), and Magsat (for coarse definitive attitude) and will include the Dynamics Explorer (DE) missions. However, whereas the computation of the TRIAD attitude matrix is very simple and straightforward, calculations of the attitude covariance matrix as currently implemented2 have been rather complicated, often requiring the computation of numerous partial derivatives as differences. To calculate these partials, the number of computations which must be performed is usually many times greater than the number required for determining the attitude. Therefore, a simple
analytical expression for the covariance matrix of the TRIAD attitude would be very useful. The derivation of such an expression is one of the goals of this paper. The greatest drawback of the TRIAD algorithm is that it can accommodate only two observations. When more than two measurements are available, these can be utilized only by cumbersomely combining the attitude solutions for the various observation-vector pairs. In addition, even when there are only two observations some accuracy is lost because part of the measurement is discarded. These two drawbacks are usually not present in an optimal algorithm, which computes a best estimate of the spacecraft attitude based on a loss function which takes into account all n measurements. Optimal algorithms, however, are usually much slower than deterministic algorithms. For the present work the chosen loss function is
(2)
Received Oct. 7, 1979; revision received April 28, 1980. Copyright © 1980 by M.D. Shuster and S.D. Oh. Published by the American Institute of Aeronautics and Astronautics with permission. * Senior Member of the Technical Staff, Attitude Systems Operation, Systems Science Division. Member AIAA. fMember of the Technical Staff, Attitude Systems Operation, Systems Science Division.
Two computationally efficient algorithms are presented for determining three-axis attitude from two or more vector observations. The first of these, the TRIAD algorithm, provides a deterministic (i.e., nonoptimal) solution for the attitude based on two vector observations. The second, the QUEST algorithm, is an optimal algorithm which determines the attitude that achieves the best weighted overlap of an arbitrary number of reference and observation vectors. Analytical expressions are given for the covariance matrices for the two algorithms using a fairly realistic model for the measurement errors. The mathematical relationship of the two algorithms and their relative merits are discussed and numerical examples are given. The advantage of computing the covariance matrix in the body frame rather than in the inertial frame (e.g., in terms of Euler angles) is emphasized. These results are valuable when a single-frame attitude must be computed frequently. They will also be useful to the mission analyst or spacecraft engineer for the evaluation of launch-window constraints or of attitude accuracies for different attitude sensor configurations.