两类惯导误差模型（PHIorPSI）（译⽂）
我亦飘零久PHI or PSI: 2 flavours of error propagation models
In order to be able to write a Kalman filter for your INS, you have to derive a linear error propagation model for your navigation system. In the existing literature, such error propagation models are broadly categorized into 2 groups.
电讯报为了使⽤卡尔曼估计惯导误差，我们需要惯导系统线性误差传播模型的推导。当前各种⽂献中使⽤的误差传播模型⼤致可以分为两类。
The first group is called as PHI-model. This is nothing but the simple perturbation of the equations of kinematics (which are wrongfully called as INS equations among navigation engineers). Becau the difference between the true attitude and the INS derived attitude is reprented as the letter PHI, this model is called as PHI.
如你所见第⼀种就是PHI⾓模型，它是在惯导的动⼒学模型基础上进⾏扰动分析得到的。因为INS的姿态和真实姿态⾓之间（P系平台系和真地理系N 之间）的误差⾓使⽤PHI⾓表⽰，因此这个模型叫做PHI⾓误差模型。
The cond group is the PSI-model. In fact, this model is also obtained as a result of a certain perturbation. However, in the PSI-model we perturb the equation of kinematics around a fictitious navigation frame which is called as the “computer frame”. The difference between the INS derived attitude and computer frame is reprented as the letter PSI. That is where this name comes from.
第⼆类误差模型成为PSI模型，也是使⽤扰动法推导⽽来。
发现的乐趣There are sufficient number of papers in the existing literature describing the derivation of psi-models. Especially the Benson’s short papers on the psi model (both his 1975 and 1978 dated papers) describe everything related with it in plain English. There are some other papers further unifying veral concepts and deriving a bunch of other models also. However, Benson’s papers (especially the one titled “A Comparison of Two Approaches to Pure-Inertial and Doppler-Inertial Error Analysis”) is all you need to learn everything regarding PSI models.
已经有许多⽂献进⾏了PSI⾓误差模型的研究，尤其是Benson的⽂章。在其⽂章中有详细的PSI⾓误差模型的推导（ “A Comparison of Two Approaches to Pure-Inertial and Doppler-Inertial Error Analysis”，DOI：10.1109/TAES.1975.308106）
Having learned both the PHI and PSI models, then you will face the real question: which model shou
ld you u? This is the main topic of this blog note.闲着
大头兰那么应该选择那种模型？
The short answer is that you should prefer the PHI model in all your Kalman filter implementations.
回答如下：
This answer may at first be surprising for you as all the canonical sources about navigation systems usually favours the psi model. So, let me elaborate my answer a little bit more.
所有关于导航系统的规范来源通常都⽀持psi模型
It is indeed true that PSI-model is more clean than the PHI model as the transport rate is not perturbed in it. Being clean means that less number of floating point calculations are required in the Kalman filter cycles. However, in today’s computing capabilities such a tinny reduction in the model computation is not important at all.
事实上PSI模型⽐PHI模型更为简洁，因为其 transport rate不受到扰动⽅程推导的影响，这意味着能够减⼩浮点计算的负载...虽然现在这⼀点⼩⼩的计算量微不⾜道了。
Even though psi-model does not have any clear advantage over phi-model, every navigation engineer must definitely learn how to derive PSI model even if he does not u it at all. (I personally had been able to learn the real meaning of navigation frame only after studying the PSI model.) Mostly becau of its conceptual importance, navigation engineers learn it in the early stages of their careers and then continue to u it out of habitual tendency. This is in fact the main reason why psi-model is more commonly ud in the navigation systems with high-grade nsors.
除臣洗马
虽然PSI模型并没有更多明显的优势，但是每⼀个导航⼯程师都需要知道PSI模型如何推导，虽然不⼀定会使⽤（我⾃⼰在学习了PSI⾓模型之后才真正理解了导航解算的框架）。主要是因为它在概念上的重要性，导航⼯程师在职业⽣涯的早期阶段就学习了它，然后出于习惯性的倾向继续使⽤它。这实际上是为什么psi模型更常⽤于带有⾼级传感器的导航系统的主要原因。
On the other hand, psi model has one big disadvantage that makes it not so suitable for low-grade units. One of the significant problems that we face in the design of low-cost systems is the azimuth initialization. High grade system can perform gyro-compasing to reduce the initial attitude uncertainty to the levels suitable for small-angle assumption. However, in low cost system, we almost always have to perform in-motion alignment starting with a large heading uncertainty. Becau of the definition of the “computer frame”, the effect of large heading uncertainty manifest itlf on the veloci
ty errors in the PSI-models. Therefore both the position and the velocity errors are affected by the non-linearity of the large attitude errors in the PSI-models. On the other hand, the large heading uncertainty only affects the position errors in the PHI-models. Therefore, PHI-models behaves better than PSI-models under large azimuth errors.
另⼀⽅⾯，psi模型有⼀个很⼤的缺点，使其不太适合低等级的IMU。在设计低成本系统时，我们⾯临的⼀个重要问题是⽅位⾓初始化。⾼精度系统可以进⾏陀螺⾃对准，使得初始失准⾓满⾜⼩⾓度假设。然⽽，在低成本系统中，我们⼏乎总是要从⼤航向误差开始进⾏动基座初始对准。由于“计算坐标系”的定义，⼤航向失准⾓对PSI模型中的速度误差的影响表现出来。因此，位置和速度误差都受到PSI模型中⼤姿态误差⾮线性的影响。另⼀⽅⾯，⼤航向不确定性仅影响PHI模型中的位置误差。因此，在⼤⽅位⾓误差下，PHI模型⽐PSI模型表现得更好。
As far as I know, in the entire literature it is only Scherzinger who us a PSI-model bad large heading filter. However, in his paper titled “Inertial Navigator Error Models For Large Heading Uncertainty”, he also recognizes the aforementioned difficulty of standard psi-models and therefore propos a modifed version of it. I find his modified psi-model unnecessarily complex. I cannot e any advantage of his method over much easier (and almost standard) method described in “T. M. Pham, Kalman filter Mechanization for INS airstart, IEEE, 1992″.
美白身体
据我所知，在整个⽂献中，只有Scherzinger使⽤了基于PSI模型的⼤航向误差⾓滤波。然⽽，在其题为“Inertial Navigator Error Models For Large Heading Uncertainty”的论⽂中，他也认识到标准psi模型的上述困难，因此提出了⼀种改进版本。我觉得他修改的psi模型不必要地复杂。我看不出他的⽅法⽐“T. M. Pham, Kalman filter Mechanization for INS airstart, IEEE, 1992”中描述的更简单（⼏乎是标准）的⽅法有任何优势。
As a result, if you are going to desing an INS with low-cost nsors, you should only consider using PHI-models as long as you do not have a robust mean of attitude initialization. You should always remember that under small angle assumption phi and psi models are equivalent. Therefore, there is absolutely no theoretical advantage of choosing one over the other. However, this does not mean that PSI-model is not important. On the contrary, if you are a navigation engineer you have to learn it by heart in order to understand the basic navigation frame concepts.
因此，如果你打算设计⼀个低成本传感器的惯导系统，只要你没有⼀个稳健的姿态初始化平均值，你就应该考虑使⽤PHI模型。您应该始终记住，在⼩⾓度假设下，PHI和PSI模型是等效的。因此，从理论上来说，选择其中⼀个绝对没有优势。然⽽，这并不意味着PSI模型不重要。相反，如果你是⼀名导航⼯程师，你必须背诵它才能理解基本的导航框架概念。
