WO2018192004A1 - 一种基于函数迭代积分的刚体姿态解算方法 - Google Patents

一种基于函数迭代积分的刚体姿态解算方法 Download PDF

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WO2018192004A1
WO2018192004A1 PCT/CN2017/082317 CN2017082317W WO2018192004A1 WO 2018192004 A1 WO2018192004 A1 WO 2018192004A1 CN 2017082317 W CN2017082317 W CN 2017082317W WO 2018192004 A1 WO2018192004 A1 WO 2018192004A1
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function
angular velocity
polynomial
rigid body
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武元新
郁文贤
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上海交通大学
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C21/00—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00
    • G01C21/10—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00 by using measurements of speed or acceleration
    • G01C21/12—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00 by using measurements of speed or acceleration executed aboard the object being navigated; Dead reckoning
    • G01C21/16—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00 by using measurements of speed or acceleration executed aboard the object being navigated; Dead reckoning by integrating acceleration or speed, i.e. inertial navigation
    • G01C21/18—Stabilised platforms, e.g. by gyroscope
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C21/00—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00
    • G01C21/10—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00 by using measurements of speed or acceleration
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C19/00—Gyroscopes; Turn-sensitive devices using vibrating masses; Turn-sensitive devices without moving masses; Measuring angular rate using gyroscopic effects
    • G01C19/56—Turn-sensitive devices using vibrating masses, e.g. vibratory angular rate sensors based on Coriolis forces
    • G01C19/5776—Signal processing not specific to any of the devices covered by groups G01C19/5607 - G01C19/5719

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  • the invention relates to the technical field of inertial navigation, robot and the like, and in particular to a rigid body attitude solving method based on function iterative integration.
  • the calculation or estimation of rigid body motion in three-dimensional space is the core problem in many fields such as physics, robotics, navigation guidance, machinery, and computer vision. Unlike the translational motions such as speed and position, the attitude cannot be directly measured and can only be obtained by indirect methods such as angular velocity integration or vector matching.
  • the attitude resolution of the angular velocity integration method is completely autonomous and does not require external information assistance, so it is favored in many applications (such as satellite navigation systems cannot function).
  • the angular velocity of the gyro output inevitably contains errors, resulting in an infinite increase in the attitude error after integration. In fact, it is generally believed that even if the angular velocity is accurate, we still cannot accurately calculate the pose due to the existence of non-commutative rotation. Rotational exchange means that different sequences of rotation will result in different postures. Therefore, an approximate method must be used in the calculation of the attitude.
  • the mainstream attitude solving method in the field of inertial navigation usually uses a quaternion to describe the current pose and a rotation vector to describe the pose increment. Under the general attitude motion, the relationship between the differential of the rotation vector and the angular velocity is more complicated, and it must be considerably simplified to apply the approximation method.
  • the mainstream attitude solving method usually uses a plurality of angular velocity or angular increment measurements (also called 'subsamples') that are continuously acquired to approximate the rotation vector, so the attitude solution can only be started when the last subsample arrives. If we only care about the solution of the pose, this is not a big problem; however, when the subsequent calculation steps need to take the pose as input, the situation is very different.
  • an object of the present invention is to provide a rigid body attitude solving method based on function iterative integration.
  • This method is based on the technique of function iterative integration, using Rodrigues vectors, Realize the reconstruction of the pose from the angular velocity analysis.
  • the differential equation form of the Rodrigue vector is simpler than the rotation vector and allows the iterative integration of polynomial functions to achieve accurate pose reconstruction.
  • the rigid body attitude solving method based on function iterative integration includes the following steps:
  • Step 1 Fit a polynomial function of the angular velocity according to the gyroscopic measurement value in the time interval;
  • Step 2 Iteratively calculates the Rodrigue vector by using the polynomial fitting function of the angular velocity and the Rodrigue vector integral equation;
  • Step 3 According to the iterative result of the Rodrigue vector, the attitude change in the time interval is given in the form of a quaternion.
  • the gyro measurement comprises an angular velocity or an angular velocity increment.
  • step 1 comprises:
  • N angular velocity measurements for time t k The angular velocity function is fitted using a polynomial that does not exceed the N-1 order; alternatively, the angular velocity function is fitted using a Chebyshev polynomial.
  • step 1 comprises:
  • N angular increment values for time t k The angular velocity function is fitted using a polynomial that does not exceed the N-1 order; alternatively, the angular velocity function is fitted using a Chebyshev polynomial.
  • step 2 comprises:
  • the angular velocity polynomial fitting function is substituted into the Rodrigue vector integral equation for iterative calculation until the convergence condition is met or the preset maximum number of iterations is reached.
  • the present invention has the following beneficial effects:
  • a new attitude solving method is proposed, which can be extended to general rigid body motion; 2) If the angular velocity is accurate, the new attitude solving method will strictly converge to the precise attitude; 3) Reconstruction from angular velocity The obtained Rodrigue vector has the form of analytic polynomial, so it can give all the attitude results in the corresponding time interval; 4) The design of the new attitude solving method does not depend on any special motion form assumptions.
  • FIG. 1 is a flowchart of a rigid body attitude solving method based on function iterative integration provided by the present invention.
  • the rigid body attitude solving method based on function iterative integration specifically includes:
  • Gyro measurements generally come in two forms: angular velocity or angular velocity increment. Discussed separately below:
  • N angular velocity measurements for time t k The angular velocity function can be fitted using a polynomial of order n (no more than N-1), ie
  • T is the uniform sampling time interval.
  • the angular velocity function can be fitted using a polynomial of order n (no more than N-1). The relationship between the angular increment value and the angular velocity is
  • the polynomial function (1) of the angular velocity can also be fitted based on the Chebyshev polynomial.
  • the attitude solving method based on function iterative integration proposed by the present invention is also applicable to other three-dimensional pose parameters, such as a rotation vector, and corresponding steps 2) and 3) are needed accordingly.
  • the adjustments are as follows:

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  • Engineering & Computer Science (AREA)
  • Radar, Positioning & Navigation (AREA)
  • Remote Sensing (AREA)
  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Automation & Control Theory (AREA)
  • Signal Processing (AREA)
  • Control Of Position, Course, Altitude, Or Attitude Of Moving Bodies (AREA)
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Abstract

一种基于函数迭代积分的刚体姿态解算方法，包括如下步骤：步骤1、根据时间区间上的陀螺测量值，拟合出角速度的多项式函数；步骤2、利用角速度的多项式拟合函数以及罗德里格向量积分方程，迭代计算罗德里格向量；步骤3、根据罗德里格向量的迭代结果，以四元数的形式给出时间区间上的姿态变化。这种方法能够拓展应用于一般的刚体运动，并且严格收敛于精确姿态。

Description

一种基于函数迭代积分的刚体姿态解算方法 技术领域
本发明涉及惯性导航、机器人等技术领域，具体而言，涉及一种基于函数迭代积分的刚体姿态解算方法。
背景技术
三维空间刚体运动的计算或估计是物理、机器人、导航制导、机械、计算机视觉等众多领域中的核心问题。与速度、位置等平移运动不同，姿态不能被直接测量，只能通过角速度积分或向量匹配等间接方式获得。角速度积分方式的姿态解算是完全自主的，不需要外部信息辅助，因此在很多(如卫星导航系统不能发挥作用的)应用场合备受青睐。然而，陀螺输出的角速度难免包含误差，造成积分后的姿态误差会无限增大。事实上，大家普遍认为：即使角速度是精确的，由于转动不可交换性的存在，我们仍然不能精确计算姿态。转动交换性是指转动的不同顺序将导致不同的姿态。因此，在姿态计算时必须采用近似的方法。
惯性导航领域的主流姿态解算方法通常采用四元数描述当前姿态、旋转向量描述姿态增量。在一般性的姿态运动下，旋转向量的微分与角速度的关系比较复杂，必须进行相当地简化才能应用近似方法来处理。主流姿态解算方法通常使用连续采集的多个角速度或角增量测量(又称‘子样’)来近似计算旋转向量，因此只能在最后一个子样到来时才能开始姿态解算。如果我们只关心姿态的解算，这不是个大问题；但是，当后续计算步骤需要将姿态作为输入时，情况就大不一样了，比如在惯性导航系统中，速度和位置的计算需要用到姿态的解算结果。另外，所有的主流姿态解算方法几乎都是以某种特殊运动(如圆锥运动)为优化指标设计的，在其他运动情形下不具有最优性。
发明内容
针对现有技术中的缺陷，本发明的目的是提供一种基于函数迭代积分的刚体姿态解算方法。本方法基于函数迭代积分的技术，利用罗德里格(Rodrigues)向量， 实现从角速度解析重建姿态。罗德里格向量的微分方程形式比旋转向量简单，且允许采用多项式函数的迭代积分实现精确的姿态重构。
根据本发明提供的基于函数迭代积分的刚体姿态解算方法，包括如下步骤：
步骤1、根据时间区间上的陀螺测量值，拟合出角速度的多项式函数；
步骤2、利用角速度的多项式拟合函数以及罗德里格向量积分方程，迭代计算罗德里格向量；
步骤3、根据罗德里格向量的迭代结果，以四元数的形式给出时间区间上的姿态变化。
优选的，所述陀螺测量值包括角速度或者角速度增量。
优选的，步骤1包括：
对于tk时刻的N个角速度测量值
Figure PCTCN2017082317-appb-000001
角速度函数采用不超过N-1阶的多项式进行拟合；或者，角速度函数采用切比雪夫多项式为基进行拟合。
优选的，步骤1包括：
对于tk时刻的N个角增量值
Figure PCTCN2017082317-appb-000002
角速度函数采用不超过N-1阶的多项式进行拟合；或者，角速度函数采用切比雪夫多项式为基进行拟合。
优选的，步骤2包括：
将角速度的多项式拟合函数代入罗德里格向量积分方程进行迭代计算，直到满足收敛条件或达到事先设定的最大迭代次数。
与现有技术相比，本发明具有如下的有益效果：
1)提出了一种新的姿态解算方法，并能够拓展应用于一般的刚体运动；2)如果角速度是精确的，新的姿态解算方法将严格收敛于精确姿态；3)从角速度重构得到的罗德里格向量具有解析多项式形式，因此可以给出相应时间区间内的所有姿态结果；4)新姿态解算方法的设计不依赖于任何特殊的运动形式假设。
附图说明
通过阅读参照以下附图对非限制性实施例所作的详细描述，本发明的其它特征、目的和优点将会变得更明显：
图1为本发明提供的基于函数迭代积分的刚体姿态解算方法的流程图。
具体实施方式
下面结合具体实施例对本发明进行详细说明。以下实施例将有助于本领域的技术人员进一步理解本发明，但不以任何形式限制本发明。应当指出的是，对本领域的普通技术人员来说，在不脱离本发明构思的前提下，还可以做出若干变化和改进。这些都属于本发明的保护范围。
结合如图1所示，本发明提供的基于函数迭代积分的刚体姿态解算方法具体包括：
1)根据时间区间[0 t]上的陀螺测量值，拟合出角速度的多项式函数；
陀螺的测量值一般有两种形式：角速度或角速度增量。下面分别讨论：
对于tk时刻的N个角速度测量值
Figure PCTCN2017082317-appb-000003
角速度函数可以采用n阶(不超过N-1)的多项式进行拟合，即
Figure PCTCN2017082317-appb-000004
其中，系数ci通过求解如下方程来确定
Figure PCTCN2017082317-appb-000005
即
Figure PCTCN2017082317-appb-000006
当n<N-1时,
Figure PCTCN2017082317-appb-000007
表示最小二乘意义下的矩阵逆,
Figure PCTCN2017082317-appb-000008
下同。其中，T为均匀采样时间间隔。
类似地,对于tk时刻的N个角增量值
Figure PCTCN2017082317-appb-000009
角速度函数可以采用n阶(不超过N-1)的多项式进行拟合。角增量值与角速度的关系为
Figure PCTCN2017082317-appb-000010
Figure PCTCN2017082317-appb-000011
即
Figure PCTCN2017082317-appb-000012
当拟合多项式的阶数较大时，
Figure PCTCN2017082317-appb-000013
或
Figure PCTCN2017082317-appb-000014
可能会面临数值计算的稳定性问题。为了提高计算的鲁棒性，角速度的多项式函数(1)也可以使用切比雪夫(Chebyshev)多项式为基进行拟合。
2)利用角速度的多项式拟合函数以及罗德里格向量积分方程，迭代计算罗德里格向量；
罗德里格向量g的迭代积分方程如下
Figure PCTCN2017082317-appb-000015
初始值g0＝0。将角速度的多项式拟合函数(1)代入上式，迭代计算，直到满足收敛条件或达到事先设定的最大迭代次数。
3)根据罗德里格向量的迭代结果，给出以时间区间开始时刻为参考的姿态四元数。
Figure PCTCN2017082317-appb-000016
上述时间区间长度t与子样个数N和均匀采样时间间隔T的关系为t＝N×T。对于更大时间区间上的姿态解算，可将其划分为若干个小时间区间，依次计算实现。
原则上，如果能接受一定程度的精度损失，本发明提出的基于函数迭代积分的姿态解算方法也适用于其他三维姿态参数，如旋转向量，此时需要对步骤2)和步骤3)做相应的调整如下：
2)利用角速度的多项式拟合函数以及旋转向量积分方程，迭代计算旋转向量；
旋转向量g的迭代积分方程如下
Figure PCTCN2017082317-appb-000017
初始值g0＝0。将角速度的多项式拟合函数(1)代入上式，迭代计算，直到满足收敛条件或达到事先设定的最大迭代次数。
3)根据旋转向量的迭代结果，给出以时间区间开始时刻为参考的姿态四元数。
Figure PCTCN2017082317-appb-000018
以上对本发明的具体实施例进行了描述。需要理解的是，本发明并不局限于上述特定实施方式，本领域技术人员可以在权利要求的范围内做出各种变化或修改，这并不影响本发明的实质内容。在不冲突的情况下，本申请的实施例和实施例中的特征可以任意相互组合。

Claims (5)

  1. 一种基于函数迭代积分的刚体姿态解算方法，其特征在于，包括如下步骤：
    步骤1、根据时间区间上的陀螺测量值，拟合出角速度的多项式函数；
    步骤2、利用角速度的多项式拟合函数以及罗德里格向量积分方程，迭代计算罗德里格向量；
    步骤3、根据罗德里格向量的迭代结果，以四元数的形式给出时间区间上的姿态变化。
  2. 根据权利要求1所述的基于函数迭代积分的刚体姿态解算方法，其特征在于，所述陀螺测量值包括角速度或者角速度增量。
  3. 根据权利要求1所述的基于函数迭代积分的刚体姿态解算方法，其特征在于，步骤1包括：
    对于tk时刻的N个角速度测量值
    Figure PCTCN2017082317-appb-100001
    k＝1，2，...N，角速度函数采用不超过N-1阶的多项式进行拟合；或者，角速度函数采用切比雪夫多项式为基进行拟合。
  4. 根据权利要求1所述的基于函数迭代积分的刚体姿态解算方法，其特征在于，步骤1包括：
    对于tk时刻的N个角增量值
    Figure PCTCN2017082317-appb-100002
    k＝1，2，...N，角速度函数采用不超过N-1阶的多项式进行拟合；或者，角速度函数采用切比雪夫多项式为基进行拟合。
  5. 根据权利要求1所述的基于函数迭代积分的刚体姿态解算方法，其特征在于，步骤2包括：
    将角速度的多项式拟合函数代入罗德里格向量积分方程进行迭代计算，直到满足收敛条件或达到事先设定的最大迭代次数。
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CN108534774B (zh) * 2018-03-21 2020-02-21 上海交通大学 基于函数迭代积分的刚体姿态解算方法及系统
CN109724597B (zh) * 2018-12-19 2021-04-02 上海交通大学 一种基于函数迭代积分的惯性导航解算方法及系统
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