四元数函数

This paper firstly reviews the history of Automated Theorem Proving and Mizar,then gives a description of methods of mechanical theorem proving and verification to mathematical proposition under Mizar system.Based on that basis,the paper has done some Mizar researches on mixed operation of quaternion,differentiati- on of special composite function and the application of Rolle\'s theorem.

本文首先介绍了定理机器证明和Mizar语言系统的发展历史，其次简单描述了Mizar系统下定理机器证明和校验数学命题的方法，在此基础上对四元数的混合运算，特殊复合函数的微分以及洛尔定理的应用等方面进行了Mizar实现。

In this paper,pose representation by quaternions are researched.With the basic quaternion algebra,the "quaternion based rotations combination theorem" is introduced and proved theoretically.On the basis of this theorem,a method for compute of pose nonliear state transfer function and its linearization is purposed.Then,the computing of the measurement function and the linearization is studied.With the Extended Kalman Filter,the pose is tracked and estimated.

研究用四元数表示物体的姿态，在四元数基本运算法则基础上提出并证明了"基于四元数的多个旋转运动合成规则"，然后将此规则为依据，推导出一种用来计算姿态非线性状态方程系数的方法，对非线性状态转移函数进行线性化，并且研究了测量方程函数的确定及线性化，借助扩展Kalman滤波实现了对姿态的跟踪。

After analyzing the relation between a quaternion function and the angular velocity it represents, a smooth angular velocity interpolation algorithm is proposed.

在分析了四元数函数与角速度的关系后，提出了角速度光滑插值的方法，并能计算出任一时间角速度的值。

The argument finds the angle of the 4- vector quaternion from the unit scalar i.e.

辐角函数可找出一4-向量四元数偏离单位纯量(即：1)之角度。

According to the clear water experiment, aeration performance of the new equipment is good with high total oxygen transfer coefficient and oxygen utilization ratio.

曝气设备的动力效率在叶轮转速为120rpm～150rpm时取得最大值，此时氧利用率和充氧能力也具有较高值。

The environmental stability of that world - including its crushing pressures and icy darkness - means that some of its most famous inhabitants have survived for eons as evolutionary throwbacks, their bodies undergoing little change.

稳定的海底环境─包括能把人压扁的压力和冰冷的黑暗─意谓海底某些最知名的栖居生物已以演化返祖的样态活了万世，形体几无变化。