1、1,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.2 空間描述和坐標(biāo)變換位置和姿態(tài)的描述,1、位置的描述 對(duì)于直角坐標(biāo)系A(chǔ)，空間任一點(diǎn)的位置可用3*1 階的列矢量 來表示(也稱位置矢量)： 除了直角坐標(biāo)系外，也可采用圓柱坐標(biāo)系或球坐標(biāo)系來描述點(diǎn)的位置。,2,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.2 空間描述和坐標(biāo)變換位置和姿態(tài)的描述,圓柱坐標(biāo)(cylindrical) : 兩個(gè)線性平移運(yùn)動(dòng)和一個(gè)旋轉(zhuǎn)運(yùn)動(dòng) 球坐標(biāo)(spherical) : 一個(gè)線性平移運(yùn)動(dòng)和兩個(gè)旋轉(zhuǎn)運(yùn)動(dòng),3,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.2 空間描述和坐標(biāo)變換位置和姿態(tài)的描述,1、位置的描述 可以引入比例因子：,比例因子可為任意值，相當(dāng)于縮放，當(dāng)為零時(shí)，表示為一個(gè)

2、長度為無窮大的向量，表示方向向量，由該向量的三個(gè)分量來表示，此時(shí)需將該向量歸一化，使長度為1。,其中：,4,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.2 空間描述和坐標(biāo)變換位置和姿態(tài)的描述,2、方位的描述 為了規(guī)定空間某剛體B的方位，另設(shè)一直角坐標(biāo)系B與此剛體固接。用坐標(biāo)系B的三個(gè)單位主矢量 ， ， 相對(duì)于坐標(biāo)系A(chǔ)的方向余弦組成的3*3 階矩陣來表示剛體B相對(duì)于A的方位：,5,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.2 空間描述和坐標(biāo)變換位置和姿態(tài)的描述,2、坐標(biāo)系在固定參考坐標(biāo)系中的表示,由表示方向的單位向量以及第四個(gè)位置向量來表示,n軸與x軸平行，o軸相對(duì)于y軸45 a軸相對(duì)于z軸45 F坐標(biāo)系位于參考坐標(biāo)系3,5,7

3、位置,例,6,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.2 空間描述和坐標(biāo)變換位置和姿態(tài)的描述,: 表示坐標(biāo)系 B主軸方向的單位矢量. : 相對(duì)于坐標(biāo)系 A的描述. 將這些單位矢量組成一個(gè) 33的矩陣，按照 的順序 . 旋轉(zhuǎn)矩陣: 標(biāo)量 可用每個(gè)矢量在其參考坐標(biāo)系中單位方向上的投影的分量來表示。,7,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.2 空間描述和坐標(biāo)變換位置和姿態(tài)的描述,3、旋轉(zhuǎn)矩陣計(jì)算 稱為旋轉(zhuǎn)矩陣，上標(biāo)A代表參考系A(chǔ)，下標(biāo)B代表被描述的坐標(biāo)系B。,重要！,8,Frame A and frame B B is rotated relative to frame A about Z by degrees,第二章 機(jī)

4、器人運(yùn)動(dòng)學(xué) 2.2 空間描述和坐標(biāo)變換位置和姿態(tài)的描述,9,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.2 空間描述和坐標(biāo)變換位置和姿態(tài)的描述,可用每個(gè)矢量在其參考坐標(biāo)系中單位方向上的投影的分量來表示: 的各個(gè)分量可用一對(duì)單位矢量的點(diǎn)積來表示 為了簡單，上式的前置上標(biāo)被省略。 由兩個(gè)單位矢量的點(diǎn)積可得到二者之間的余弦，因此可以理解為什么旋轉(zhuǎn)矩陣的各分量常被稱作為方向余弦。components of rotation matrices are often referred to as direction cosines,PAPB=|PA|PB|cos,10,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.2 空間描述和坐標(biāo)變換位置和姿

5、態(tài)的描述,進(jìn)一步觀察 ，可以看出矩陣的行是單位矢量 A在 B中的描述. 因?yàn)?為坐標(biāo)系A(chǔ)相對(duì)于 B的描述 由轉(zhuǎn)置得到 這表明旋轉(zhuǎn)矩陣的逆矩陣等于它的轉(zhuǎn)置,11,4、旋轉(zhuǎn)矩陣性質(zhì) 1） 矩陣有9個(gè)元素，其中只有3個(gè)是獨(dú)立的。因?yàn)槿齻€(gè)列矢量都是單位主矢量，且兩兩相互垂直，所以它的9個(gè)元素滿足6個(gè)約束條件（正交條件）：,2） 把矢量在B中的坐標(biāo)表達(dá)式變?yōu)樵贏中的坐標(biāo)表達(dá)式的變換矩陣：,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.2 空間描述和坐標(biāo)變換位置和姿態(tài)的描述,3） 是正交矩陣，即有：,12,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.2 空間描述和坐標(biāo)變換坐標(biāo)系的描述,用 和 來描述坐標(biāo)系,13,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.3

6、映射坐標(biāo)變換,1、平移坐標(biāo)系的映射 設(shè)坐標(biāo)系B與A具有相同的方位，但是B的坐標(biāo)原點(diǎn)與A不重合，用位置矢量 描述它相對(duì)于A的位置，稱為B相對(duì)于A的平移矢量。如果點(diǎn)P在坐標(biāo)系B中的位置為 ，則它相對(duì)于坐標(biāo)系A(chǔ)的位置矢量 可由矢量相加得出：,14,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.3 映射坐標(biāo)變換,2、旋轉(zhuǎn)坐標(biāo)系的映射 設(shè)坐標(biāo)系B和A有共同的原點(diǎn)，但是兩者的方位不同。 同一點(diǎn)P在兩個(gè)坐標(biāo)系A(chǔ)和B中的描述 和 具有以下變換關(guān)系 ，稱為坐標(biāo)系旋轉(zhuǎn)方程。 用旋轉(zhuǎn)矩陣 表示坐標(biāo)系B相對(duì) 于A的方位。同樣，用 描述坐標(biāo)系 A相對(duì)于B的方位。二者都是正交矩 陣，兩者互逆。,15,Example: Frame B is

7、rotated relative to frame A about Z by 30 degrees. Here Z is pointing out of the page. Writing the unit vectors of B in terms of A and stacking them as the columns of the rotation matrix: The original vector P is not changed, we compute a new description relative to another frame.,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.3 映射坐

8、標(biāo)變換,16,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.3 映射坐標(biāo)變換,關(guān)于一般坐標(biāo)系的映射 坐標(biāo)系B的原點(diǎn)與A的既不重合，方位也不相同。 復(fù)合變換是由坐標(biāo)旋轉(zhuǎn)和坐標(biāo)平移共同作用的。,17,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.3 映射坐標(biāo)變換,齊次變換 復(fù)合變換式對(duì)于點(diǎn) 而言是非齊次的，但是可以將其表示成等價(jià)的齊次變換形式： 其中，41的列向量表示三維空間的點(diǎn)，稱為點(diǎn)的齊次坐標(biāo)，仍然記為 或 。上式可以寫成矩陣形式： 齊次變換矩陣也代表坐標(biāo)平移與坐標(biāo)旋轉(zhuǎn)的復(fù)合,可將其分解成兩個(gè)矩陣相乘的形式：,18,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.3 映射坐標(biāo)變換,連續(xù)旋轉(zhuǎn)平移變換 連續(xù)相對(duì)轉(zhuǎn)動(dòng)，可把基本矩陣連乘起來，由于選轉(zhuǎn)矩陣不可交換

9、，故完成轉(zhuǎn)動(dòng)的次序是重要的。 如果B坐標(biāo)系相對(duì)于A坐標(biāo)系的坐標(biāo)軸轉(zhuǎn)動(dòng)，則對(duì)旋轉(zhuǎn)矩陣左乘相應(yīng)的基本旋轉(zhuǎn)矩陣，如果B坐標(biāo)系相對(duì)于B坐標(biāo)系的坐標(biāo)軸轉(zhuǎn)動(dòng)，則對(duì)旋轉(zhuǎn)矩陣右乘相應(yīng)的基本旋轉(zhuǎn)矩陣。 例：假設(shè)B相對(duì)A的軸依次進(jìn)行了下面三個(gè)變換： 1）繞x軸旋轉(zhuǎn) 度； 2）接著平移 ； 3）最后繞y軸旋轉(zhuǎn) 度。,19,Example: Frame B is rotated relative to frame A about Z by 30 degrees, translated 10 units in , and translated 5 unit in . Find , where . The definit

10、ion of frame B is We use the definition of B just given a transformation:,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.3 映射坐標(biāo)變換,20,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.4 算子: 平移、旋轉(zhuǎn)和變換,用于坐標(biāo)系間點(diǎn)的映射的通用數(shù)學(xué)表達(dá)式被稱為算子包括點(diǎn)的平移算子、矢量旋轉(zhuǎn)算子和平移加旋轉(zhuǎn)算子。 1) 平移算子（Translational operators） A translation moves a point in space a finite distance along a given vector direction. Only

11、one coordinate system need be involved. It turns out that translating the point in space is accomplished with the same mathematics as mapping the point to a second frame. The distinction is: when a vector is moved “forward” relative to a frame, we may consider either that the vector moved forward or

12、 that the frame moved backword. The mathematics involved in the two cases is identical, only our view of the situation is different.,21,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.4 算子: 平移、旋轉(zhuǎn)和變換,運(yùn)算的結(jié)果得到一個(gè)新的矢量，計(jì)算如下： 用矩陣算子寫出平移變換 where q is the signed magnitude of the translation along the vector direction .,22,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.4 算子: 平移

13、、旋轉(zhuǎn)和變換,算子 可以被看成是一種特殊形式的齊次變換: 式中 是平移矢量 Q 的分量 通過定義B相對(duì)于A的位置， (用 ) , 我們使得這兩個(gè)描述具有相同的數(shù)學(xué)表達(dá)式?，F(xiàn)在引入了 ，我們可以用它來描述坐標(biāo)系和映射。,23,2) 旋轉(zhuǎn)算子（Rotational operators） Another interpretation of a rotation matrix is as a rotational operator that operates on a vector and changes that vector to a new vector, , by means of a rot

14、ation, R. When a rotation matrix is shown as an operator, no sub- or superscripts appear, because it is not viewed as relating two frame. We may write: Again, the mathematics is the same, only our interpretation is different. How to obtain rotational matrices that are to be used as operators: The ro

15、tation matrix that rotates vectors through some rotation, R, is the same as the rotation matrix that describes a frame rotated by R relative to the refrence frame.,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.4 算子: 平移、旋轉(zhuǎn)和變換,24,Although a rotation matrix is easily viewed as an operator, we can also define another notation for a rot

16、ational operator that clearly indicates which axis is being rotated about: is a rotational operator that performs a rotation about the axis direction by degrees. For example:,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.4 算子: 平移、旋轉(zhuǎn)和變換,25,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.4 算子: 平移、旋轉(zhuǎn)和變換,Example: Figure shows a vector . We wish to compute the vector obt

17、ained by rotating this vector about Z by 30 degrees. Call the new vector . The rotation matrix that rotates vectors by 30 degrees about Z is the same as the rotation matrix that describes a frame rotated 30 degrees about Z relative to the reference frame. Thus, the correct rotational operator is,26,

18、第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.4 算子: 平移、旋轉(zhuǎn)和變換,3) 變換算子（Transformation operators） As with vectors and rotation matrices, a frame has another interpretation as a transformation operator. In the interpretation, only one coordinate system is involved, and so the symbol T is used without sub- or superscripts. How to obtain

19、 homogeneous transform that are to be used as operators: The transform that rotates by R and translated by Q is the same as the transform that describes a frame rotated by R and translated by Q relative to the refrence frame.,27,Example: Figure shows vector . We wish to rotate it about Z by 30 degre

20、es and translate it 10 units in and 5 units in . Find ,where . The operator T, which performs the translation and rotation:,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.4 算子: 平移、旋轉(zhuǎn)和變換,28,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.5 總結(jié)和說明,Summary of interpretations (1) 齊次變換陣是坐標(biāo)系的描述. describes the frame B relative to the frame A. （description of a frame） (2)齊次變換

21、陣是變換映射. maps .（） (3)齊次變換陣是變換算子. T operates on to create . From this point on, the terms frame and transform will both be used to refer to a position vector plus an orientation. Frame is the term favored in speaking of a description, Transform is used most frequently when function as a mapping or ope

22、rator is implied. Note that transformation are generalizations of (and subsume) translations and rotations; we will often use the term transform when speaking of a pure rotation (or translation).,29,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.6 變換算法,齊次變換的計(jì)算 1）相乘：對(duì)于給定的坐標(biāo)系A(chǔ)、B和C： 2）求逆：如果知道坐標(biāo)系B相對(duì)A的描述，希望得到A相對(duì)B的描述：,30,Example: Frame B

23、is rotated relative to frame A about by 30 degrees and translated four units in and three units in . Thus, we have a description of . Find . The frame defining B is:,第二章 機(jī)器人運(yùn)動(dòng)學(xué) 2.6 變換算法,31,CHAPTER 2: Spatial description 2.7 變換方程,Figure indicates a situation in which a frame D can be expressed as pro

24、ducts of transformations in two different ways: We can set these two descriptions of equal to construct a transform equation: Transform equations can be used to solve for transforms in the case of n unknown transforms and n transform equations.,32,Consider in the case that all transforms are known e

25、xcept . Here, we have one transform equation and one unknown transform, hence, we easily find its solution: 注意：在所有的途中，我們都采用了坐標(biāo)系的圖形表示法，即用一個(gè)坐標(biāo)系的原點(diǎn)指向另一個(gè)坐標(biāo)系的原點(diǎn)的箭頭來表示。將箭頭串聯(lián)起來，通過簡單的變換方程就可得到混合坐標(biāo)系。箭頭的方向指明了坐標(biāo)系定義的方式。如果有一個(gè)箭頭的方向與串聯(lián)的方向相反，就先求出它的逆 。,CHAPTER 2: Spatial description 2.7 變換方程,33,Example: 假定已知操作臂末端執(zhí)行器的

26、坐標(biāo)系 , 它是相對(duì)于操作臂基座的坐標(biāo)系B定義的，又已知工作臺(tái)相對(duì)于操作臂基座的空間位置 , 并且已知工作臺(tái)上螺栓的坐標(biāo)系相對(duì)于工作臺(tái)坐標(biāo)系的位置 計(jì)算螺栓相對(duì)于操作手的位姿：,CHAPTER 2: Spatial description 2.7 變換方程,34,CHAPTER 2: Spatial description 2.8 姿態(tài)的其它描述方法,Problem: 能否用少于九個(gè)數(shù)字來表示一個(gè)姿態(tài)? A result from linear algebra (known as Cayleys formula): for any proper orthonormal matrix R, th

27、ere exists a skew-symmetric matrix (S=-ST) S such that: a skew-symmetric matrix of dimension 3 is specified by three parameters as: 任何 33的旋轉(zhuǎn)矩陣都可用三個(gè)參量確定.,35,顯然，旋轉(zhuǎn)矩陣的九個(gè)分量線性相關(guān)。實(shí)際上，對(duì)于一個(gè)旋轉(zhuǎn)矩陣R很容易 寫出六個(gè)線性無關(guān)的分量。假定R為三列： These three vectors are the unit axes of some frame writtern in terms of the refrence fram

28、e. Each is a unit vector, and all three must be mutually perpendicular, so we see that there are six constrains on the nine parameters: 是否能找到一種姿態(tài)表示法，用三個(gè)參量就能簡便進(jìn)行表達(dá)?,CHAPTER 2: Spatial description 2.8 姿態(tài)的其它描述方法,36,Whereas translations along three mutually perpendicular axes are quite easy to visualize

29、, rotations seem less intuitive. Unfortunately people have a hard time describing and specifying orientation in three-dimensional space. One difficulty is that rotations dont generally commute. That is: Example: 考慮兩個(gè)軸旋轉(zhuǎn)，一個(gè)繞Z轉(zhuǎn)30度，另一個(gè)繞X軸轉(zhuǎn)30度。 :,CHAPTER 2: Spatial description 2.8 姿態(tài)的其它描述方法,37,Example:固

30、連在坐標(biāo)系B上的點(diǎn) （1）繞z軸旋轉(zhuǎn)90度: （1）繞z軸旋轉(zhuǎn)90度； （2）然后繞y軸轉(zhuǎn)90度； （2）再平移4，-3，7； （3）最后再平移4，-3，7。 （3）然后繞y軸轉(zhuǎn)90度。,CHAPTER 2: Spatial description 2.8 姿態(tài)的其它描述方法,38,1) X-Y-Z 固定角坐標(biāo)系（fixed angles） 下面介紹描述坐標(biāo)系B姿態(tài)的另一種方法: Start with the frame coincident with a known refrence frame A. Rotate B first about by an angle , then about

31、 by an angle , and, finally, about by an angle . 每個(gè)旋轉(zhuǎn)都是繞著固定參考坐標(biāo)系A(chǔ)的軸。我們規(guī)定這種姿態(tài)的表示法為X-Y-Z固定角坐標(biāo)系?！肮潭ā币辉~是指旋轉(zhuǎn)是在固定（即不運(yùn)動(dòng)的）參考坐標(biāo)系中確定的。有時(shí)把它們定義為回轉(zhuǎn)角、俯仰角和偏轉(zhuǎn)角。,CHAPTER 2: Spatial description 2.8 姿態(tài)的其它描述方法,39,CHAPTER 2: Spatial description 2.8 姿態(tài)的其它描述方法,可以直接推導(dǎo)等價(jià)旋轉(zhuǎn)矩陣，因?yàn)樗械男D(zhuǎn)都是繞著參考坐標(biāo)系各軸的， where is shorthand for , for

32、 . 最重要的是搞清楚上式中的旋轉(zhuǎn)順序. Equation above is correct only for rotations performed in the order: about by an angle , then about by an angle , and, finally, about by an angle . 常常使人感興趣的是逆解問題，即從一個(gè)旋轉(zhuǎn)矩陣等價(jià)推出X-Y-Z固定角坐標(biāo)系。逆解取決于求解一組超越方程；如果方程相當(dāng)于一個(gè)已知的旋轉(zhuǎn)矩陣，那么就有九個(gè)方程和三個(gè)未知量。在這九個(gè)方程中有六個(gè)方程是相關(guān)的。,40,CHAPTER 2: Spatial descrip

33、tion 2.8 姿態(tài)的其它描述方法,Let: In summary: Although a second solution exists, by using the positive square root in the formula for , we always compute the single solution for which . This is usually a good practice. If , the solution degenerates. In those cases, one possible convention is to choose .,41,2)

34、 Z-Y-X 歐拉角（Euler angles） 坐標(biāo)系 B的另一種表示法如下: Start with the frame coincident with a known refrence frame A. Rotate B first about by an angle , then about by an angle , and, finally, about by an angle . In this representation, each rotation is performed about an axis of the moving system B rather than on

35、e of the fixed refrence A. Such sets of three rotations are called Euler angles. Note that each rotations takes place about an axis whose location depends upon the preceding rotations.,CHAPTER 2: Spatial description 2.8 姿態(tài)的其它描述方法,42,We can write: 注意這個(gè)結(jié)果與以相反順序繞固定軸旋轉(zhuǎn)三次得到的結(jié)果完全相同！總之，這是一個(gè)不太直觀的結(jié)果：三次繞固定軸旋轉(zhuǎn)

36、的最終姿態(tài)和以相反順序三次繞運(yùn)動(dòng)坐標(biāo)軸旋轉(zhuǎn)的最終姿態(tài)相同。 因?yàn)榈葍r(jià)，所以無需通過旋轉(zhuǎn)矩陣的反復(fù)計(jì)算去求Z-Y-X的歐拉角。.,CHAPTER 2: Spatial description 2.8 姿態(tài)的其它描述方法,43,3) Z-Y-Z Euler angles Describing the orientation of a frame B as follow: Start with the frame coincident with a known refrence frame A. Rotate B first about by an angle , then about by an

37、angle , and, finally, about by an angle . Extracting:,CHAPTER 2: Spatial description 2.8 姿態(tài)的其它描述方法,44,4) 其它角坐標(biāo)系的表示法 In the preceding subsections we have seen three conventions for specifying orientation: X-Y-Z fixed angles, Z-Y-X Euler angles, and Z-Y-Z Euler angles. 每個(gè)表示法均需要按一定順序進(jìn)行三次繞主軸的旋轉(zhuǎn)。這些表示法是24

38、種表示法中的典型方法，且都被稱作角坐標(biāo)系表示法。其中，12種為固定角坐標(biāo)系法，另12種為歐拉角坐標(biāo)系法。注意到由于二者之間的對(duì)偶性，對(duì)于繞主軸連續(xù)旋轉(zhuǎn)的旋轉(zhuǎn)矩陣實(shí)際上只有12種唯一的參數(shù)表示方法。 感興趣的同學(xué)可以參考本書附錄B,CHAPTER 2: Spatial description 2.8 姿態(tài)的其它描述方法,45,CHAPTER 2: Spatial description 2.8 姿態(tài)的其它描述方法,5) 等效軸角坐標(biāo)系表示法 With the notation we give the description of an orientation by giving an axis,

39、 X, and an angle 30 degrees. This is an example of an equivalent angle-axis representation. If the axis is a general direction (rather than one of the unit directions) any orientation may be obtained through proper axis and angle selection. Describing the orientation of a frame B as follow: Start wi

40、th the frame coincident with a known refrence frame A. Then Rotate B first about the vector by an angle according to the right-hand rule. Vector is called the equivalent axis of a finite rotation.,46,CHAPTER 2: Spatial description 2.8 姿態(tài)的其它描述方法,A general orientation of B relative to A may be written as or . The specification of the vector requires only two parameters, because its length is always taken to be one. The angle specifies a third parameter. The equivalent rotation matrix is: where , and . The sign of is determined by the right-hand rule, with the thumb pointing along