机器人学第二章运动学.ppt

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1、1,第二章 机器人运动学2.2 空间描述和坐标变换位置和姿态的描述,1、位置的描述 对于直角坐标系A，空间任一点的位置可用3*1 阶的列矢量 来表示(也称位置矢量)：除了直角坐标系外，也可采用圆柱坐标系或球坐标系来描述点的位置。,2,第二章 机器人运动学2.2 空间描述和坐标变换位置和姿态的描述,圆柱坐标(cylindrical):两个线性平移运动和一个旋转运动 球坐标(spherical):一个线性平移运动和两个旋转运动,3,第二章 机器人运动学2.2 空间描述和坐标变换位置和姿态的描述,1、位置的描述 可以引入比例因子：,比例因子可为任意值，相当于缩放，当为零时，表示为一个长度为无穷大的向

2、量，表示方向向量，由该向量的三个分量来表示，此时需将该向量归一化，使长度为1。,其中：,4,第二章 机器人运动学2.2 空间描述和坐标变换位置和姿态的描述,2、方位的描述 为了规定空间某刚体B的方位，另设一直角坐标系B与此刚体固接。用坐标系B的三个单位主矢量，相对于坐标系A的方向余弦组成的3*3 阶矩阵来表示刚体B相对于A的方位：,5,第二章 机器人运动学2.2 空间描述和坐标变换位置和姿态的描述,2、坐标系在固定参考坐标系中的表示,由表示方向的单位向量以及第四个位置向量来表示,n轴与x轴平行，o轴相对于y轴45a轴相对于z轴45F坐标系位于参考坐标系3,5,7位置,例,6,第二章 机器人运动

3、学2.2 空间描述和坐标变换位置和姿态的描述,:表示坐标系 B主轴方向的单位矢量.:相对于坐标系 A的描述.将这些单位矢量组成一个 33的矩阵，按照的顺序.旋转矩阵:标量 可用每个矢量在其参考坐标系中单位方向上的投影的分量来表示。,7,第二章 机器人运动学2.2 空间描述和坐标变换位置和姿态的描述,3、旋转矩阵计算 称为旋转矩阵，上标A代表参考系A，下标B代表被描述的坐标系B。,重要！,8,Frame A and frame B B is rotated relative to frame A about Z by degrees,第二章 机器人运动学2.2 空间描述和坐标变换位置和姿态的描述

4、,9,第二章 机器人运动学2.2 空间描述和坐标变换位置和姿态的描述,可用每个矢量在其参考坐标系中单位方向上的投影的分量来表示:的各个分量可用一对单位矢量的点积来表示 为了简单，上式的前置上标被省略。由两个单位矢量的点积可得到二者之间的余弦，因此可以理解为什么旋转矩阵的各分量常被称作为方向余弦。components of rotation matrices are often referred to as direction cosines,PAPB=|PA|PB|cos,10,第二章 机器人运动学2.2 空间描述和坐标变换位置和姿态的描述,进一步观察，可以看出矩阵的行是单位矢量 A在 B中的

5、描述.因为 为坐标系A相对于 B的描述 由转置得到这表明旋转矩阵的逆矩阵等于它的转置,11,4、旋转矩阵性质 1）矩阵有9个元素，其中只有3个是独立的。因为三个列矢量都是单位主矢量，且两两相互垂直，所以它的9个元素满足6个约束条件（正交条件）：,2）把矢量在B中的坐标表达式变为在A中的坐标表达式的变换矩阵：,第二章 机器人运动学2.2 空间描述和坐标变换位置和姿态的描述,3）是正交矩阵，即有：,12,第二章 机器人运动学2.2 空间描述和坐标变换坐标系的描述,用 和 来描述坐标系,13,第二章 机器人运动学2.3 映射坐标变换,1、平移坐标系的映射 设坐标系B与A具有相同的方位，但是B的坐标原

6、点与A不重合，用位置矢量 描述它相对于A的位置，称为B相对于A的平移矢量。如果点P在坐标系B中的位置为，则它相对于坐标系A的位置矢量 可由矢量相加得出：,14,第二章 机器人运动学2.3 映射坐标变换,2、旋转坐标系的映射 设坐标系B和A有共同的原点，但是两者的方位不同。同一点P在两个坐标系A和B中的描述 和 具有以下变换关系，称为坐标系旋转方程。用旋转矩阵 表示坐标系B相对 于A的方位。同样，用 描述坐标系 A相对于B的方位。二者都是正交矩 阵，两者互逆。,15,Example:Frame B is rotated relative to frame A about Z by 30 degr

7、ees.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 机器人运动学2.3 映射坐标变换,16,第二章 机器人运动学2.3 映射坐标变换,关于一般坐标系的映射坐标系B的原点与A的既不重

8、合，方位也不相同。复合变换是由坐标旋转和坐标平移共同作用的。,17,第二章 机器人运动学2.3 映射坐标变换,齐次变换 复合变换式对于点 而言是非齐次的，但是可以将其表示成等价的齐次变换形式：其中，41的列向量表示三维空间的点，称为点的齐次坐标，仍然记为 或。上式可以写成矩阵形式：齐次变换矩阵也代表坐标平移与坐标旋转的复合,可将其分解成两个矩阵相乘的形式：,18,第二章 机器人运动学2.3 映射坐标变换,连续旋转平移变换 连续相对转动，可把基本矩阵连乘起来，由于选转矩阵不可交换，故完成转动的次序是重要的。如果B坐标系相对于A坐标系的坐标轴转动，则对旋转矩阵左乘相应的基本旋转矩阵，如果B坐标系相

9、对于B坐标系的坐标轴转动，则对旋转矩阵右乘相应的基本旋转矩阵。例：假设B相对A的轴依次进行了下面三个变换：1）绕x轴旋转 度；2）接着平移；3）最后绕y轴旋转 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 definition of frame B is We use the definition of B just given a transformation:,第

10、二章 机器人运动学2.3 映射坐标变换,20,第二章 机器人运动学2.4 算子:平移、旋转和变换,用于坐标系间点的映射的通用数学表达式被称为算子包括点的平移算子、矢量旋转算子和平移加旋转算子。1)平移算子（Translational operators）A translation moves a point in space a finite distance along a given vector direction.Only one coordinate system need be involved.It turns out that translating the point in

11、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 that the frame moved backword.The mathematics involved in the two cases is identical,on

12、ly our view of the situation is different.,21,第二章 机器人运动学2.4 算子:平移、旋转和变换,运算的结果得到一个新的矢量，计算如下：用矩阵算子写出平移变换 where q is the signed magnitude of the translation along the vector direction.,22,第二章 机器人运动学2.4 算子:平移、旋转和变换,算子 可以被看成是一种特殊形式的齐次变换:式中 是平移矢量 Q 的分量 通过定义B相对于A的位置，(用),我们使得这两个描述具有相同的数学表达式。现在引入了，我们可以用它来描述坐

13、标系和映射。,23,2)旋转算子（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 rotation,R.When a rotation matrix is shown as an operator,no sub-or superscripts appear,because it is not view

14、ed 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 rotation matrix that rotates vectors through some rotation,R,is the same as the rotation matrix that describes a frame r

15、otated by R relative to the refrence frame.,第二章 机器人运动学2.4 算子:平移、旋转和变换,24,Although a rotation matrix is easily viewed as an operator,we can also define another notation for a rotational operator that clearly indicates which axis is being rotated about:is a rotational operator that performs a rotation

16、 about the axis direction by degrees.For example:,第二章 机器人运动学2.4 算子:平移、旋转和变换,25,第二章 机器人运动学2.4 算子:平移、旋转和变换,Example:Figure shows a vector.We wish to compute the vector obtained by rotating this vector about Z by 30 degrees.Call the new vector.The rotation matrix that rotates vectors by 30 degrees about

17、 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,第二章 机器人运动学2.4 算子:平移、旋转和变换,3)变换算子（Transformation operators）As with vectors and rotation matrices,a frame has another interpretation as a t

18、ransformation operator.In the interpretation,only one coordinate system is involved,and so the symbol T is used without sub-or superscripts.How to obtain 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 descr

19、ibes 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 degrees and translate it 10 units in and 5 units in.Find,where.The operator T,which performs the translation and rotation:,第二章 机器人运动学2.4 算子:平移、旋转和变换,28,第二章

20、机器人运动学2.5 总结和说明,Summary of interpretations(1)齐次变换阵是坐标系的描述.describes the frame B relative to the frame A.（description of a frame）(2)齐次变换阵是变换映射.maps.（）(3)齐次变换阵是变换算子.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 orientat

21、ion.Frame is the term favored in speaking of a description,Transform is used most frequently when function as a mapping or operator 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(

22、or translation).,29,第二章 机器人运动学2.6 变换算法,齐次变换的计算 1）相乘：对于给定的坐标系A、B和C：2）求逆：如果知道坐标系B相对A的描述，希望得到A相对B的描述：,30,Example:Frame B 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 机器人运动学2.6 变换算法,31,C

23、HAPTER 2:Spatial description2.7 变换方程,Figure indicates a situation in which a frame D can be expressed as products 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

24、of n unknown transforms and n transform equations.,32,Consider in the case that all transforms are known except.Here,we have one transform equation and one unknown transform,hence,we easily find its solution:注意：在所有的途中，我们都采用了坐标系的图形表示法，即用一个坐标系的原点指向另一个坐标系的原点的箭头来表示。将箭头串联起来，通过简单的变换方程就可得到混合坐标系。箭头的方向指明了坐标系

25、定义的方式。如果有一个箭头的方向与串联的方向相反，就先求出它的逆。,CHAPTER 2:Spatial description2.7 变换方程,33,Example:假定已知操作臂末端执行器的坐标系,它是相对于操作臂基座的坐标系B定义的，又已知工作台相对于操作臂基座的空间位置,并且已知工作台上螺栓的坐标系相对于工作台坐标系的位置 计算螺栓相对于操作手的位姿：,CHAPTER 2:Spatial description2.7 变换方程,34,CHAPTER 2:Spatial description2.8 姿态的其它描述方法,Problem:能否用少于九个数字来表示一个姿态?A result f

26、rom linear algebra(known as Cayleys formula):for any proper orthonormal matrix R,there 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的旋转矩阵都可用三个参量确定.,35,显然，旋转矩阵的九个分量线性相关。实际上，对于一个旋转矩阵R很容易 写出六个线性无关的分量。假定R为三列：These three

27、vectors are the unit axes of some frame writtern in terms of the refrence frame.Each is a unit vector,and all three must be mutually perpendicular,so we see that there are six constrains on the nine parameters:是否能找到一种姿态表示法，用三个参量就能简便进行表达?,CHAPTER 2:Spatial description2.8 姿态的其它描述方法,36,Whereas translat

28、ions along three mutually perpendicular axes are quite easy to visualize,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:考虑两个轴旋转，一个绕Z转30度，另一个绕X

29、轴转30度。:,CHAPTER 2:Spatial description2.8 姿态的其它描述方法,37,Example:固连在坐标系B上的点（1）绕z轴旋转90度:（1）绕z轴旋转90度；（2）然后绕y轴转90度；（2）再平移4，-3，7；（3）最后再平移4，-3，7。（3）然后绕y轴转90度。,CHAPTER 2:Spatial description2.8 姿态的其它描述方法,38,1)X-Y-Z 固定角坐标系（fixed angles）下面介绍描述坐标系B姿态的另一种方法:Start with the frame coincident with a known refrence fr

30、ame A.Rotate B first about by an angle,then about by an angle,and,finally,about by an angle.每个旋转都是绕着固定参考坐标系A的轴。我们规定这种姿态的表示法为X-Y-Z固定角坐标系。“固定”一词是指旋转是在固定（即不运动的）参考坐标系中确定的。有时把它们定义为回转角、俯仰角和偏转角。,CHAPTER 2:Spatial description2.8 姿态的其它描述方法,39,CHAPTER 2:Spatial description2.8 姿态的其它描述方法,可以直接推导等价旋转矩阵，因为所有的旋转都是绕

31、着参考坐标系各轴的，where is shorthand for,for.最重要的是搞清楚上式中的旋转顺序.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.常常使人感兴趣的是逆解问题，即从一个旋转矩阵等价推出X-Y-Z固定角坐标系。逆解取决于求解一组超越方程；如果方程相当于一个已知的旋转矩阵，那么就有九个方程和三个未知量。在这九个方程中有六个方程是相关的。,40,CH

32、APTER 2:Spatial description2.8 姿态的其它描述方法,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 choo

33、se.,41,2)Z-Y-X 欧拉角（Euler angles）坐标系 B的另一种表示法如下: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 one of

34、 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 description2.8 姿态的其它描述方法,42,We can write:注意这个结果与以相反顺序绕固定轴旋转三次得到的结果完全相同！总之，这是一个不太直观的结果：三次绕固定轴旋转的最终姿态和以相反

35、顺序三次绕运动坐标轴旋转的最终姿态相同。因为等价，所以无需通过旋转矩阵的反复计算去求Z-Y-X的欧拉角。.,CHAPTER 2:Spatial description2.8 姿态的其它描述方法,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 angle,and,finally

36、,about by an angle.Extracting:,CHAPTER 2:Spatial description2.8 姿态的其它描述方法,44,4)其它角坐标系的表示法 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.每个表示法均需要按一定顺序进行三次绕主轴的旋转。这些表示法是24种表示法中的典型方法，且都被称作角坐标系表示法。其中，12种

37、为固定角坐标系法，另12种为欧拉角坐标系法。注意到由于二者之间的对偶性，对于绕主轴连续旋转的旋转矩阵实际上只有12种唯一的参数表示方法。感兴趣的同学可以参考本书附录B,CHAPTER 2:Spatial description2.8 姿态的其它描述方法,45,CHAPTER 2:Spatial description2.8 姿态的其它描述方法,5)等效轴角坐标系表示法 With the notation we give the description of an orientation by giving an axis,X,and an angle 30 degrees.This is an

38、 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 with the frame coincident with a known refrenc

39、e 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 description2.8 姿态的其它描述方法,A general orientation of B relative to A may be written as or.The specification of the vector requires

40、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 the positive sense of.The inverse problem:,47,Example:A Frame B is describ

41、ed as initially coincident with A.We then rotate B about the vector(passing through the origin)by an amount degrees.Give the frame description of B.The frame description of B is:Up to this point,all rotations we have discussed have been about axes that pass through the origin of the reference system

42、.If we encounter a problem for which this is not true,we can reduce the problem to the“axis through the origin”case by defining additional frames whose origins lie on the axis and then solving a transform equation.,CHAPTER 2:Spatial description2.8 姿态的其它描述方法,48,Example:A Frame B is described as initi

43、ally coincident with A.We then rotate B about the vector(passing through the point)by an amount degrees.Give the frame description of B.Before the rotation,A and B are coincident.We difine two new frames,A and B,which are coincident with each other and have the same orientation as A and B respective

44、ly,but are translated relative to A by an offset that places their origins on the axis of rotation.We will choose:Description of B in terms of B:,CHAPTER 2:Spatial description2.8 姿态的其它描述方法,49,CHAPTER 2:Spatial description习题,圆柱坐标(cylindrical):两个线性平移运动和一个旋转运动 球坐标(spherical):一个线性平移运动和两个旋转运动,50,CHAPTER 2:Spatial description习题,根据下图,give the value of.,