机器人学中的旋量理论与几何：教程

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"Geometry and Screw Theory for Robotics" 是一本关于机器人学中的几何与螺旋理论教程，由 Stefano Stramigioli 和 Herman Bruyninckx 于2001年撰写。该教程强调了在机器人领域中，基于螺旋几何和代数的方法相较于其他技术展现出的优势。本书深入探讨了刚体运动、配置、速度、力和动力学等方面的概念。章节1首先介绍了刚体的运动，从欧几里得空间的基本概念出发，如点、向量、标量积、距离测量、垂直性和向量积，这些是理解机器人运动的基础。作者将刚体的配置描述分为两类：利用Lie群（特别提到SE(3)）的方法，以及有限扭（finite twist）描述，这有助于简化复杂系统的分析。接着，章节转向了刚体的速度，即twists，它们作为SE(3)元素被定义，并与螺旋理论相结合，展示了其在描述刚体旋转和平移速度上的便利性。Wrenches（力矩）也是关键概念，它们同样作为SE(3)中的元素出现，用于表示作用在刚体上的外力和力偶，其在螺旋理论中的应用使得力的解析更加直观。动力学部分着重于动能、惯性变换以及欧拉方程的讨论，这些都是设计和控制机器人系统时必不可少的物理基础。书中还涵盖了Lie代数的指数映射，特别是SO(3)和SE(3)元素的指数映射，这对于解析刚体运动的动态行为至关重要。章节2进一步扩展到串行机构的配置动力学，包括刚体链的运动学，通过这些概念，可以理解和分析复杂机械结构的行为，比如关节空间和驱动物理的关联。总结来说，"Geometry and Screw Theory for Robotics" 提供了一个系统性的框架，帮助读者掌握如何运用螺旋理论这一强大的工具来处理机器人学中的各种问题，从单个刚体的运动控制到多关节机械臂的运动分析，都是本书涵盖的重要知识点。通过学习和应用这些理论，工程师们能够提升设计和优化机器人系统的能力。

1.3. CONFIGURATIONS OF A RIGID BODY

is therefore almost identical to the two-dimensional case, but now we are dealing

with a 3 × 3 matrix.

Remark 2 It is important to note that:

• The columns of R

are the unit vectors along the axes of Ψ

, expressed in the coordi-

nate system Ψ

• The rows of R

are the the unit vectors along the axes of Ψ

, expressed in the coor-

dinate system Ψ

A direct consequence of this is that the inverse of R

, i.e., R

, is equal to the transpose

of R

. This property holds for any rotation matrix R:

−1

= R

The following identities are useful properties when rotating any vectors v, w ∈ R

R(v ∧ w)=(Rv) ∧ (Rw), (1.8)

and

R ˜wR



(Rw), (1.9)

where the “tilde” operator is deﬁned in Eq. (1.1).

As an example of a rotation in E(3), consider a change of coordinates due to a rotation

of θ around ˆy

. The matrix representing this change of coordinates looks like:





cos(θ)0sin(θ)

010

−sin(θ)0cos(θ)





It is possible to see that the set of matrices satisfying R

−1

= R

is a three-dimensional

matrix Lie group, which is called the orthonormal group and indicated with O(3). First,

it fulﬁlls all properties of a group:

• Associativity: R

∈ O(3) ⇒ (R

= R

• Identity: I ∈ O(3).

• Inverse: R ∈ O(3) ⇒ R

−1

∈ O(3),RR

−1

= I.

Moreover, it is a Lie group, because the composition operation is continuous. Since

R = I, ∀R ∈ O(3), and since the determinant of a product is the product of the

determinants, the determinant of any matrix in O(3) must be ±1. This shows that O(3)

is composed of two disjoint components, one whose matrices have determinant equal to

−1, and one consisting of the matrices with determinant equal to +1. The former is not

a group by itself, because the product of two such matrices has determinant +1. The

latter is again a Lie group, called the special orthonormal group, and denoted by SO(3):

SO(3) = {R ∈ R

3×3

; R

−1

= R

, det R =1}.

Remark 3 Note that the previous rotations are all around a speciﬁc point in space, namely

the common origin of the coordinate systems. This implies that SO(3) can only be used

to describe rotations around a speciﬁc point and not any rotation! This is not explicitly

mentioned in most of the robotics literature.

1. MOTION OF A RIGID BODY

The Lie group SO(3) is the appropriate mathematical model of the orientations of a rigid

body. Also the angular velocities of rigid bodies must get a good mathematical model.

This is given by the so-called Lie algebra of SO(3), which is denoted with so(3). A Lie

algebra describes the local (“ﬁrst order”) properties of its parent Lie group (irrespective

of what the Lie group represent), while maintaining the geometric properties.

From the theory on left and right invariant vector ﬁelds, Eq. (B.5) and Eq. (B.6), we can

see that elements of the algebra should look like T

= R

−1

R, or like T

−1

, for any

curve R(t) ∈ SO(3). Because R

−1

= R

, differentiating the equation R

R = I yields:

R + R

R =0⇒ R

R = −(R

which means that the matrix T

is skew-symmetric. Similarly, T

is also skew-symmetric,

so that so(3) is the vector space of skew-symmetric matrices. Note that due to the struc-

ture of the matrix commutator of the algebra, the commutation of skew-symmetric ma-

trices is still skew-symmetric.

There is a bijective relation between 3 × 3 skew-symmetric matrices and 3 vectors as it

was shown in Eq. (1.1).

It is straightforward to see that the algebra matrix commutator of so(3) corresponds to

the vector product of the corresponding three vectors:

[˜ω

, ˜ω



∧ ω

∀ω

,ω

∈ R

The elements of so(3) correspond to the angular velocity of the frames whose relative

change of coordinates is represented by the matrix R

∈ SO(3). We will analyse in more

detail the general case, i.e., angular and translational velocities. A lot of coordinate rep-

resentations of rotations exists, besides the above-mentioned skew-symmetric matrix

representation of SO(3); for example, the unit quaternions, which are isomorphic to an-

other Lie group, namely SU(2) (the Special Unitary group. They are both double covers

of SO(3) (i.e., each rotation in SO(3) corresponds to two elements in the double cover

group). In contrast to SO(3), they are both topologically speaking simply connected.

SO(3) is instead not contractible and it is a typical example of a non simply connected

manifold “without holes.” This means that if we deﬁne a continuous closed curve in

SO(3) (a ring), there is not always a way to continuously make this ring smaller and

smaller in order to let it become a point.

Remark 4 (Vector product) It is a misconception to interpret the vector product in three

space as an extra structure on E

∗

(the scalar product being the other structure). This

is NOT the case, because the vector product is a consequence of the fact that for E the

Lie group of rotations SO(3) can be intrinsically deﬁned. As we have just seen, the Lie

algebra so(3) has a commutator deﬁned on it and therefore, after a choice of orientation

of space, the intrinsic bijection between so(3) and E

∗

carries the commutation operation

deﬁned in so(3) over to the vector product in E

∗

It is possible to ﬁnd two bijections in the following way. A vector of E

∗

is characterized by

a direction d, an orientation v and a module m. An element of so(3) represents an angular

velocity. It is possible to see that any angular motion leaves a line d



invariant. This is

the ﬁrst step in this bijection: the vector of E

∗

which we associate to a vector in so(3) will

have as direction the line left invariant by this rotation (d = d



). We still need a module

and an orientation. It is possible to show that there exists a positive deﬁnite metric on

so(3), called the Killing form (Selig 1996), which deﬁnes the magnitude of a rotation in a

unique, coordinate-independent way. We can therefore choose as m the module of this

angular velocity vector calculated using the metric on so(3). The only choice we are left

1.3. CONFIGURATIONS OF A RIGID BODY

ˆx

ˆy

Figure 1.4: A general change of coordinates.

with is that of the orientation. We clearly have two possible choices to orient the line. If

we look at the rotation motion around its axis as a clockwise motion, we can orient the

line as going away from us or as coming toward us. In the ﬁrst case we say that we

choose a right-handed orientation and in the second case a left-handed orientation. This

is clearly related to the orientation chosen for the Euclidean space because if we consider

a point x moving due to the described rotation, we can consider a vector x(t) − o and a

vector x(t + dt) − o where o is the orthogonal projection

of x on the axis of rotation. The

direction of the vector (x(t) − o) ∧ (x(t + dt) −o) is the associated direction of rotation.

Notice that if we look at this motion through a mirror, the orientation is changed because,

under the same rotational motion, a line oriented toward the mirror will be seen through

the mirror as a line oriented in the opposite direction. This is called a reﬂection.

Translations

Translations are much simpler than rotations. If we have two coordinate systems dis-

placed with respect to each other, the change of coordinates is simple:

=(Ψ

o Ψ

−1

)(p

)=p

+ p

, (1.10)

where p

represents the position of the origin of Ψ

with respect to the origin of Ψ

, and

expressed as a numerical vector in one of the two frames (either one, because they are

not rotated with respect to each other). Obviously this set of translations is also a Lie

group, due to the well-known additivity of translation vectors.

General changes of coordinates

It is now possible to consider a general transformation including any rotation and/or

translation. Figure 1.4 shows the change of coordinates from a 2D frame Ψ

to another

frame Ψ

. p

∈ R

contains the coordinates of the vector (p − o

) ∈E

∗

expressed in the

frame Ψ

. Taking the coordinates of the following vector equality

(p − o

)=(p − o

)+(o

− o

yields

= R

+ p

We can consider an orthogonal projection since we are in a Euclidean space

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机器人学中的旋量理论与几何：教程

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