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标题“ROS imu_filter_madgwick 功能包涉及到的文档”和描述“在ros当中常用的一个功能包，主要是对imu 的数据进行一个处理”所涉及的知识点主要集中在ROS（Robot Operating System）平台上的一个特定功能包imu_filter_madgwick的介绍及其应用。这个功能包基于一个公开发表的学术论文，该论文主要介绍了一个新颖的基于梯度下降算法的姿态估计方法。下面将详细展开介绍这些知识点。 ROS是一个用于机器人应用开发的元操作系统，它提供了硬件抽象描述、底层设备控制、常用功能实现、进程间消息传递和包管理等机制。ROS的一个特点就是将软件模块化，这使得它在社区中得到了广泛使用，并形成了丰富的功能包集合。ROS中常用的功能包，例如imu_filter_madgwick，是用于处理传感器数据，特别是惯性测量单元（IMU）数据的功能模块。 IMU是一种传感器设备，集成了加速度计、陀螺仪和磁力计，常被用来确定和维持物体的空间定位，例如无人机（UAV）、机器人和虚拟现实（VR）头盔等设备。在ROS中，imu_filter_madgwick就是用来对IMU数据进行滤波处理，从而得到更加准确的姿态估计。在文档提及的论文《Estimation of IMU and MARG orientation using a gradient descent algorithm》中，Sebastian O.H. Madgwick，Andrew J.L. Harrison和Ravi Vaidyanathan描述了一种新的基于梯度下降算法的姿态估计算法。该算法能够有效地支持计算高效的可穿戴惯性人体运动跟踪系统，适合于康复应用。该算法能够处理包括三轴陀螺仪和加速度计在内的惯性测量单元（IMUs），以及同时包括三轴磁力计的磁角速率和重力（MARG）传感器阵列。MARG实现还包含了磁干扰补偿。该算法使用了四元数表示法，允许加速度计和磁力计数据能够被用于一种经过分析推导和优化的梯度下降算法中，用于计算陀螺仪测量误差的方向作为一个四元数导数。实验通过使用一个商业可用的方向传感器，并利用光学测量系统获取的参考方向测量数据，对性能进行了实证评估。性能也与专有的基于Kalman滤波的姿态传感器算法进行了基准测试。结果表明，该算法在静态和动态条件下的精度与基于Kalman的算法相当，静态RMS误差小于0.8度，动态RMS误差小于1.7度。算法低的计算负载和低采样率下的操作能力显著减少了可穿戴惯性运动跟踪所需的硬件和电力，使得创建轻质、廉价的系统成为可能，这些系统能够长时间运行。在介绍中，文章强调了准确的方向测量在多个领域中的关键作用，包括航空航天、机器人技术、导航、人类运动分析以及与机器的互动。在康复领域，运动跟踪是至关重要的使能技术，尤其是在临床环境外监测病人的活动，理想情况下可以连续监测病人的活动，并随后进行矫正。此外，由于对给定文件的“【部分内容】”中的文字进行了OCR扫描，存在一些识别错误和遗漏，因此在解读具体细节时需要结合上下文进行合理推断，以保证知识点的准确性和完整性。这要求对ROS及相关的算法原理有深入理解，从而能够准确阐释功能包imu_filter_madgwick的实际应用和技术优势。

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Estimation of IMU and MARG orientation using a

gradient descent algorithm

Sebastian O.H. Madgwick, Andrew J.L. Harrison, Ravi Vaidyanathan

Abstract—This paper presents a novel orientation algorithm

designed to support a computationally efﬁcient, wearable inertial

human motion tracking system for rehabilitation applications. It

is applicable to inertial measurement units (IMUs) consisting of

tri-axis gyroscopes and accelerometers, and magnetic angular

rate and gravity (MARG) sensor arrays that also include tri-axis

magnetometers. The MARG implementation incorporates mag-

netic distortion compensation. The algorithm uses a quaternion

representation, allowing accelerometer and magnetometer data to

be used in an analytically derived and optimised gradient descent

algorithm to compute the direction of the gyroscope measurement

error as a quaternion derivative. Performance has been evaluated

empirically using a commercially available orientation sensor

and reference measurements of orientation obtained using an

optical measurement system. Performance was also benchmarked

against the propriety Kalman-based algorithm of orientation

sensor. Results indicate the algorithm achieves levels of accuracy

matching that of the Kalman based algorithm; < 0.8

◦

static

RMS error, < 1.7

◦

dynamic RMS error. The implications of the

low computational load and ability to operate at small sampling

rates signiﬁcantly reduces the hardware and power necessary

for wearable inertial movement tracking, enabling the creation

of lightweight, inexpensive systems capable of functioning for

extended periods of time.

I. INTRODUCTION

The accurate measurement of orientation plays a critical

role in a range of ﬁelds including: aerospace [1], robotics [2],

[3], navigation [4], [5] and human motion analysis [6], [7]

and machine interaction [8]. In rehabilitation, motion tracking

is vital enabling technology, in particular for monitoring

outside clinical environs; ideally, a patient’s activities could

be continuously monitored, and subsequently corrected. While

extensive work has been performed for motion tracking for

rehabilitation, an unobtrusive, wearable system capable of

logging data for extended periods of time has yet to be

realized. Existing systems often require a laptop or handheld

PC to be carried by the subject due to the processing, data

storage, and power requirements of the sensory equipment.

This is not practical outside of a laboratory environment, thus

detailed data may only be obtained for short periods of time

for a limited range of subject’s motion. More precise data

representative of a subject’s natural behaviour over extended

periods of time (e.g. an entire day or even a week) would be

Sebastian Madgwick is with the Department of Mechanical Engineering,

University of Bristol, e-mail: s.madgwick@bristol.ac.uk.

Ravi Vaidyanathan is with the Department of Mechanical Engineering, Uni-

versity of Bristol, Queens Building, BS8 1TR and the Department of Sys-

tems Engineering at the US Naval Postgraduate School, Monterey, CA, USA,

93940. e-mail: r.vaidyanathan@bristol.ac.uk.

Andrew Harrison is with the Department of Mechanical Engineering, Uni-

versity of Bristol, e-mail: andrew.j.l.harrison@bristol.ac.uk.

of signiﬁcant utility in this realm. In a recent survey, Zhoua

[7], cited real time operation, wireless properties, correctness

of data, and portability as major deﬁciencies that must be

addressed to realize a clinically viable system.

A. Inertial Motion Tracking Systems

Whilst a variety of technologies enable the measurement of

orientation, inertial based sensory systems have the advantage

of being completely self contained such that the measurement

entity is constrained neither in motion nor to any speciﬁc

environment or location. An IMU (Inertial Measurement Unit)

consists of gyroscopes and accelerometers enabling the track-

ing of rotational and translational movements. In order to

measure in three dimensions, tri-axis sensors consisting of 3

mutually orthogonal sensitive axes are required. A MARG

(Magnetic, Angular Rate, and Gravity) sensor is a hybrid

IMU which incorporates a tri-axis magnetometer. An IMU

alone can only measure an attitude relative to the direction of

gravity which is sufﬁcient for many applications [2], [1], [6].

MARG systems, also known as AHRS (Attitude and Heading

Reference Systems) are able to provide a complete measure-

ment of orientation relative to the direction of gravity and the

earth’s magnetic ﬁeld. An orientation estimation algorithm is

a fundamental component of any IMU or MARG system. It is

required to fuse together the separate sensor data into a single,

optimal estimate of orientation.

The Kalman ﬁlter [9] has become the accepted basis for

the majority of orientation algorithms [2], [10], [11], [12]

and commercial inertial orientation sensors; xsens [13], micro-

strain [14], VectorNav [15], Intersense [16], PNI [17] and

Crossbow [18] all produce systems founded on its use. The

widespread use of Kalman-based solutions are a testament

to their accuracy and effectiveness, however, they have a

number of disadvantages. They can be complicated to im-

plement which is reﬂected by the numerous solutions seen

in the subject literature [2], [10], [11], [12], [19], [20], [21],

[22], [23]. The linear regression iterations, fundamental to the

Kalman process, demand sampling rates which can far exceed

the subject bandwidth (e.g. a sampling rate between 512 Hz

[13] and 30 kHz [14] may be necessary for human motion

capture applications where system portability is critical). The

state relationships describing rotational kinematics in three-

dimensions typically require large state vectors and an ex-

tended Kalman ﬁlter implementation [2], [12], [19] to linearise

the problem.

These challenges demand a large computational load for

implementation of Kalman-based solutions and provide a clear

2011 IEEE International Conference on Rehabilitation Robotics

Rehab Week Zurich, ETH Zurich Science Cit

, Switzerland, June 29 - Jul

1, 2011

motivation for alternative approaches. Previous approaches

to address these issues have implemented either fuzzy pro-

cessing [1], [3] or frequency domain ﬁlters [24] to favour

accelerometer measurements of orientation at low angular

velocities and the integrated gyroscope measurements at high

angular velocities. Such an approach is simple but may only

be effective under limited operating conditions. Bachman et al

[25] and Mahony et al [26] present separate algorithms which

both employ a complementary ﬁlter process. This algorithm

structure has been shown to provide effective performance at

relatively little computational expense.

This paper introduces orientation estimation algorithm that

is applicable to both IMU and MARG systems. The algorithm

employs a quaternion representation of orientation (as in:

[25], [12], [19]) to describe the coupled nature of orientations

in three-dimensions and is not subject to the problematic

singularities associated with an Euler angle representation.

A complete derivation and empirical evaluation of the new

algorithm is presented. Its performance is benchmarked against

existing commercial ﬁlters and veriﬁed with optical tracking

system.

II. O

RGANISATION OF PAPER

Section III delineates the mathematical derivation of the

orientation estimation algorithm, including a description of the

parameterization and compensation for magnetic distortion.

Section IV describes the experimental equipment used to test

and verify the performance of the algorithm. Section V quanti-

ﬁes the experimental testing and accuracy of the algorithm and

compares it to existing systems. Section VII expands brieﬂy

gives details on implementations of the system underway

currently in our laboratory in human motion tracking while

Section VI summarizes conclusions and contributions of this

work. Throughout the paper, a notation system of leading

superscripts and subscripts adopted from Craig [27] is used

to denote the relative frames of orientations and vectors. A

leading subscript denotes the frame being described and a

leading superscript denotes the frame this is with reference

to. For example, For example,

q describes the orientation of

frame B relative to frame A and

v is a vector described in

frame A.

III. A

LGORITHM DERIVATION

A. Orientation from angular rate

A tri-axis gyroscope will measure the angular rate about

the x, y and z axes of the senor frame, termed ω

, ω

and

respectively. If these parameters (in rads

−1

) are arranged

into the vector

ω deﬁned by equation (1), the quaternion

derivative describing rate of change of the earth frame relative

to the sensor frame

q can be calculated [28] as equation (2).

The ⊗ operate denotes a quaternion product and the ˆ accent

denotes a normalised vector of unit length.

ω =



0 ω



(1)

q =

q ⊗

ω (2)

The orientation of the earth frame relative to the sensor

frame at time t,

ω,t

, can be computed by numerically

integrating the quaternion derivative

ω,t

as described by

equations (3) and (4), provided that initial conditions are

known. In these equations,

is the angular rate measured

at time t, Δt is the sampling period and

est,t−1

is the

previous estimate of orientation. The subscript ω indicates that

the quaternion is calculated from angular rates.

ω,t

est,t−1

⊗

(3)

ω,t

est,t−1

ω,t

Δt (4)

B. Orientation from a homogenous ﬁeld

In the context of an orientation estimation algorithm, it will

initially be assumed that an accelerometer will measure only

gravity and a magnetometer will measure only the earth’s

magnetic ﬁeld. 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.

A quaternion representation requires a single solution to be

found. This may be achieved through the formulation of an

optimisation problem where an orientation of the sensor,

is found as that which aligns a predeﬁned reference direction

of the ﬁeld in the earth frame,

d, with the measured ﬁeld

in the sensor frame,

s; thus solving (5) where equation (6)

deﬁnes the objective function.

min

q∈

s) (5)

s)=

∗

⊗

d ⊗

q −

s (6)

Many optimisation algorithms exist but the gradient descent

algorithm is one of the simplest to both implement and

compute. Equation (7) describes the gradient descent algorithm

for n iterations resulting in an orientation estimation of

n+1

based on an ‘initial guess’ orientation

and a variable

step-size μ. Equation (8) computes an error direction on the

solution surface deﬁned by the objective function, f , and its

Jacobian, J .

k+1

− μ

∇f(



∇f(



,k=0, 1, 2...n (7)

∇f(

s)=J

(

d)f(

s) (8)

Equations (7) and (8) describe the general form of the

algorithm applicable to a ﬁeld predeﬁned in any direction.

However, if the reference direction of the ﬁeld is deﬁned to

only have components within 1 or 2 of the principle axis of

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