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"蘇黎世聯邦理工學院機器人技術講義：動力學控制與逆向運動解matlab程序碼"

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The Robotics Technology Lecture Notes from the Swiss Federal Institute of Technology in Zurich cover topics such as kinematics, dynamics, inverse kinematics, and MATLAB code for motion analysis. These lecture notes are designed for engineers who are interested in robotics technology. The kinematics section covers the study of motion, including the position and orientation of robotic systems. It explains how to represent positions using Cartesian coordinates and Euler angles, and how to transform between Euler angles and rotation matrices. This section provides a foundation for understanding the movement of robots in various environments. The dynamics section delves into the forces and torques that govern the motion of robotic systems. It includes topics such as joint-space dynamic control, where the focus is on controlling the motion of individual joints, and task-space dynamic control, which looks at controlling the overall motion of the robot. The lecture notes also discuss inverse dynamics for floating-base systems, such as humanoid robots and legged robots, and quasi-static control for robots like robot dogs that require balance control. The MATLAB code provided in the lecture notes allows engineers to implement the concepts discussed in the lectures and analyze the motion of robotic systems. This hands-on approach helps reinforce the theoretical concepts and provides practical experience in robotics technology. Overall, the Robotics Technology Lecture Notes from the Swiss Federal Institute of Technology in Zurich offer a comprehensive overview of kinematics, dynamics, and control in robotics. Whether you are a beginner or an experienced engineer in the field, these lecture notes provide valuable insights into the world of robotics technology.

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Figure 2.3: A passive rotation can be interpreted as the rotation of a coordinate frame,

and an active rotation as the rotation of an object u which yields v.

2.4.2 Active vs. Passive Rotation

Rotations can have two different interpretations, which lead to the deﬁnition of the

so-called active and passive rotations.

Passive Rotation

Passive rotations, also known as rotation transformations, correspond to a mapping

between coordinate frames as shown in (2.21). A passive rotation C

maps the same

object u from frame B to frame A:

u = C

u (2.24)

Active Rotation

An active rotation, often indicated with a 3 × 3 matrix R, is an operator that rotates a

vector

u to a vector

v in the same reference frame A:

v = R ·

u. (2.25)

Active rotations are not very relevant in robot dynamics and are hence not used in the

course of this lecture.

2.4.3 Elementary Rotations

The most simple and at the same time most often appearing rotations are elementary

rotations, i.e. rotations around one of the basis vectors e

, e

or e

. Given a rotation

angle ϕ, the three elementary rotations are:

= e

= C

(ϕ) =





1 0 0

0 cos ϕ −sin ϕ

0 sin ϕ cos ϕ





(2.26)

= e

= C

(ϕ) =





cos ϕ 0 sin ϕ

0 1 0

−sin ϕ 0 cos ϕ





(2.27)

= e

= C

(ϕ) =





cos ϕ −sin ϕ 0

sin ϕ cos ϕ 0

0 0 1





(2.28)

2.4.4 Composition of Rotations

Consider three reference frames A, B and C. The coordinates of vector u can be

mapped from B to A by writing

u = C

u. (2.29)

We can also write

u = C

u. (2.30)

By combining the last two equations, we can write

u = C

· (C

= C

(2.31)

The resulting rotation matrix C

= C

· C

can be interpreted as the rotation

obtained by rotating frame A until it coincides with frame B, and then rotating frame

B until it coincides with frame C.

2.4.5 Representation of Rotations

As discussed in the previous sections, generic rotations in the three-dimensional space

are represented by 3 × 3 rotation matrices, i.e. by means of 9 parameters. These

parameters, however, are not independent, but constrained by the orthogonality condi-

tions shown in (2.23). Hence, only three independent parameters such as Euler angles

are needed to obtain a minimal representation of rotations in space. Other non-minimal

representations can be derived, namely the angle-axis and the unit quaternion represen-

tation. We will brieﬂy discuss advantages and disadvantages of each parameterization,

as well as derive the mapping from one implementation to the other. For a more de-

tailed analysis of three dimensional rotations (which goes beyond the scope of this

lecture), the reader is referred to [2].

Euler Angles

A rotation in space can be understood as a sequence of three elementary rotations

deﬁned in (2.26) to (2.28). To fully describe all possible orientations, two successive

rotations should not be made around parallel axes. When the ﬁrst and third rotations are

made around the same axis, the parameterization is called proper Euler angles. When

all three angles are different, we typically refer to Tait–Bryan, Cardan or roll-pitch-yaw

angles. The latter ones are often used in robotics.

Note: Please note again that there applies a different type of algebra to rotations

than what we know from typical position vectors. Hence, never add, subtract or simply

multiply Euler angles, angle-axis, or quaternions.

Figure 2.4: ZYZ Euler Angles as three successive rotations around z, y, and z axes.

Rotation from A-frame to D-frame: (z–y’–z”) – (yaw–pitch–yaw)

ZYZ Euler Angles ZYZ Euler Angles are also known as proper Euler Angles. The

rotation angles can be collected in a parameter vector

R,eulerZY Z









. (2.32)

The resulting rotation matrix is obtained by a concatenation of elementary rotations

(see Fig.2.4) given by

= C

(y)C

) ⇒

r = C

ADD





cos z

−sin z

sin z

cos z

0 0 1









cos y 0 sin y

0 1 0

−sin y 0 cos y









cos z

−sin z

sin z

cos z

0 0 1









− s

−c

− c

+ c

− c

−c





(2.33)

Analyzing (2.33) allows to ﬁnd the solution of the inverse problem. Given a rotation

matrix









, (2.34)

the ZYZ Euler angles are given by

R,eulerZY Z















atan2 (c

, c

)

atan2



+ c

, c



atan2 (c

, −c

)







. (2.35)

Figure 2.5: ZXZ Euler Angles as three successive rotations around z, x, and z axes.

Rotation from A-frame to D-frame: (z–x’–z”) – (yaw–roll–yaw)

ZXZ Euler Angles ZXZ Euler Angles are also known as proper Euler Angles. The

rotation angles can be collected in a parameter vector

R,eulerZXZ









. (2.36)

The resulting rotation matrix is obtained by a concatenation of elementary rotations

(Fig.2.5) given by

= C

(x)C

) ⇒

r = C

ADD





cos z

−sin z

sin z

cos z

0 0 1









1 0 0

0 cos x −sin x

0 sin x cos x









cos z

−sin z

sin z

cos z

0 0 1









− c

−c

− c

+ c

− s

−c





(2.37)

Analyzing (2.37) allows to ﬁnd the solution of the inverse problem. Given a rotation

matrix









, (2.38)

the Euler Angles are given by

R,eulerZXZ















atan2 (c

, −c

)

atan2



+ c

, c



atan2 (c

, c

)







. (2.39)

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在省略号处填写一行或多行代码，完成如下功能。同时，在考生文件夹下有个文件 data . txt ，其中记录了2019年 QS 全球大学排校信息，示例如下： 1，麻省理工学院，美国 2，斯坦福大学，美国 3，哈佛大学，美国第1列为排名，第2列为学校名称，第3列为学校所属的国家，字段之间用’，’隔开。程序读取 data . txt 文件内容，统计出现的国家个数以及每个国家上榜大学的数量及名称输出结果格式示例如下：英国：5：牛津大学剑桥大学帝国理工学院伦敦大学学院爱丁堡大学瑞士：1：苏黎世联邦理工学院 f = open (' data . txt ',' r ') for line in f : …… for d in unis : print ({:>4):(:>4):()'. format (——）

以下是代码实现： ``` country_dict = {} with open('data.txt', 'r') as f: for line in f: line = line.strip() rank, uni, country = line.split(',') ...美国 3 麻省理工学院斯坦福大学哈佛大学 ```

cubli平衡方块资料

Cubli是一种由瑞士苏黎世联邦理工学院开发的平衡方块，它具有令人印象深刻的自平衡和行走能力。Cubli的外观像一个小正方体，由三个相互垂直的动力滚轮组成。滚轮通过电机驱动，并且可以在不同方向上旋转。 Cubli的...

在省略号处填写一行或多行代码，完成时，在考生文件夹下有个文件 data . txt ，其中记录了2019年 QS 全球大学排校信息，示例如下：+ 1，麻省理工学院，美国 2，斯坦福大学，美国 3，哈佛大学，美国第1列为排名，第2列为学校名称，第3列为学校所属的国家，字段之间用程序读取 data . txt 文件内容，统计出现的国家个数以及每个国家上榜大学的数出结果格式示例如下：英国：5：牛津大学剑桥大学帝国理工学院伦敦大学学院爱丁堡大学＋瑞士：1：苏黎世联邦理工学院 f = open (' data . txt ',' r ') for line in f : …… for d in unis : print ({:>4):(:>4):()'. format (——）

```python country_dict = {} with open('data.txt', 'r') as f: for line in f: line = line.strip() if line: rank, name, country = line.split('，') country_dict[country] = country_dict.get(country, ...

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