Madgwick Orientation Filter算法在IMU与MARG传感器中的应用
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更新于2024-07-15
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"这两篇论文由Sebastian O. H. Madgwick撰写，分别名为《An efficient orientation filter for inertial and inertial/magnetic sensor arrays》和《Estimation of IMU and MARG orientation using a gradient descent algorithm》，主要探讨了惯性传感器阵列（IMU）和磁力/惯性传感器阵列（MARG）的姿态解算问题。"
在这两篇论文中，Madgwick提出了一个针对IMU和MARG传感器的新颖姿态滤波器，该滤波器尤其关注了计算效率、低采样率下的有效性以及自适应参数调整。在第一篇论文中，Madgwick集中讨论了包含三轴陀螺仪和加速度计的IMU系统，而在第二篇论文中，他进一步将三轴磁力计纳入考虑，形成MARG传感器阵列，处理磁场畸变和陀螺仪偏置漂移的问题。
滤波器的核心是四元数表示法，这是一个数学工具，用于描述三维空间中的旋转。Madgwick使用四元数来融合加速度计和磁力计数据，通过分析推导出的梯度下降算法计算出陀螺仪测量误差的方向。这种方法的优势在于：
1. 计算成本低：每次滤波更新仅需进行109次（IMU）或277次（MARG）标量运算，这在硬件资源有限的嵌入式系统中具有显著优势。
2. 在低采样率下表现良好：即使在10Hz这样的低采样率下，也能有效地提供稳定且准确的姿态解算结果。
3. 参数自适应：滤波器包含1个（IMU）或2个（MARG）可调参数，这些参数可以根据系统的实际观测特性进行定义，增强了滤波器对不同环境和设备的适应性。
论文通过实验评估了该滤波器的性能，使用了商业化的定向传感器，并与光学测量系统获取的参考姿态数据进行了对比，验证了其在实际应用中的优秀性能和准确性。
Madgwick的这两篇工作对惯性导航领域有重要贡献，它们提出的梯度下降算法为实时姿态解算提供了一个高效且实用的解决方案，特别是在无人机、机器人和穿戴设备等需要实时、精确姿态信息的场景中，这种滤波技术有着广泛的应用前景。
If the direction of an earth’s field is known in the earth frame, a measurement of the
field’s direction within the sensor frame will allow an orientation of the sensor frame relative
to the earth frame to be calculated. However, for any given measurement there will not be
a unique sensor orientation solution, instead there will infinite solutions represented by all
those orientations achieved by the rotation the true orientation around an axis parallel with
the field. In some applications it may be acceptable to use an Euler angle representation
allowing an incomplete solution to be found as two known Euler angles and one unknown [5];
the unknown angle being the rotation around an axis parallel with the direction of the field.
A quaternion representation requires a complete solution to be found. This may be achieved
through the formulation of an optimisation problem where an orientation of the sensor,
S
E
ˆ
q,
is that which aligns a predefined reference direction of the field in the earth frame,
E
ˆ
d,
with the measured direction of the field in the sensor frame,
S
ˆ
s, using the rotation operation
described by equation (5). Therefore
S
E
ˆ
q may be found as the solution to (14) where equation
(15) defines the objective function. The components of each vector are defined in equations
(16) to (18).
min
S
E
ˆ
q∈<
4
f(
S
E
ˆ
q,
E
ˆ
d,
S
ˆ
s) (14)
f(
S
E
ˆ
q,
E
ˆ
d,
S
ˆ
s) =
S
E
ˆ
q
∗
⊗
E
ˆ
d ⊗
S
E
ˆ
q −
S
ˆ
s (15)
S
E
ˆ
q =
q
1
q
2
q
3
q
4
(16)
E
ˆ
d =
0 d
x
d
y
d
z
(17)
S
ˆ
s =
0 s
x
s
y
s
z
(18)
Many optimisation algorithms exist but the gradient descent algorithm is one of the
simplest to both implement and compute. Equations (19) describes the gradient descent
algorithm for n iterations resulting in an orientation estimation of
S
E
ˆ
q
n+1
based on an ‘initial
guess’ orientation
S
E
ˆ
q
0
and a step-size µ. Equation (20) computes the gradient of the solution
surface defined by the objective function and its Jacobian; simplified to the 3 row vectors
defined by equations (21) and (22) respectively.
S
E
q
k+1
=
S
E
ˆ
q
k
− µ
∇f(
S
E
ˆ
q
k
,
E
ˆ
d,
S
ˆ
s)
∇f(
S
E
ˆ
q
k
,
E
ˆ
d,
S
ˆ
s)
, k = 0, 1, 2...n (19)
∇f(
S
E
ˆ
q
k
,
E
ˆ
d,
S
ˆ
s) = J
T
(
S
E
ˆ
q
k
,
E
ˆ
d)f(
S
E
ˆ
q
k
,
E
ˆ
d,
S
ˆ
s) (20)
f(
S
E
ˆ
q
k
,
E
ˆ
d,
S
ˆ
s) =
2d
x
(
1
2
− q
2
3
− q
2
4
) + 2d
y
(q
1
q
4
+ q
2
q
3
)+
2d
x
(q
2
q
3
− q
1
q
4
) + 2d
y
(
1
2
− q
2
2
− q
2
4
)+
2d
x
(q
1
q
3
+ q
2
q
4
) + 2d
y
(q
3
q
4
− q
1
q
2
)+
2d
z
(q
2
q
4
− q
1
q
3
) − s
x
2d
z
(q
1
q
2
+ q
3
q
4
) − s
y
2d
z
(
1
2
− q
2
2
− q
2
3
) − s
z
(21)
7
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