z
z
o
a n o x World Reference Frame y x Joint Reference Frame y
x Tool Reference Frame
y
【机器人的参考坐标系】
a
n
1. 机器人运动学的矩阵表示
Representation of a Point in Space

A point P in space can be represented by its three coordinates relative to a reference frame as:
nx n F y nz 0
ox oy oz 0
ax ay az 0
px py pz 1
【齐次变换矩阵】
3. 变换的表示 Representation of Transformations
当空间的坐标系（向量、物体或运动坐标系）相对于固定的参考坐 标系运动时，这一运动可以用类似于表示坐标系的方式来表示。

n o 0 n a 0 ao 0
(the dot-product of n and o vectors must be zero)

n 1
(the magnitude of the length of the vector must be 1) and
o 1
a 1

The same can be achieved by:
3.

Pre-multiply by each matrix:
p1, xyz =Rot ( x, ) pnoa
p2, xyz Trans(l1, l2 , l3 ) p1, xyz Trans(l1, l2 , l3 ) Rot ( x, ) pnoa
pxyz p3, xyz Rot ( y, ) p2, xyz Rot ( y, ) Trans(l1, l2 , l3 ) Rot ( x, ) pnoa


In this case, matrices representing each transformation are post-multiplied. If transformations are relative to both the Universe frame and the current frame, each matrix is accordingly multiplied, either pre- or post-.
【相对于旋转坐标系（当前坐标系/运动坐标系）的变换】
4. 变换矩阵的逆
Inverse of Matrices

The following steps must be taken to calculate the inverse of a matrix:




Calculate the determinant of the matrix. Transpose the matrix. Replace each element of the transposed matrix by its own minor (adjoint matrix). Divide the converted matrix by the determinant.
Fnew = Trans (dx ,dy ,dz )
1 0 0 0 0 1 0 0 0 d x nx n 0 dy y 1 d z nz 0 1 0 ox oy o z 0
F
ax ay az 0
old
Fnew
p x nx n py y p z nz 1 0
no a
上式包含了正确的右手法则关系，所以一般使用这个等式判断3个向量之间的关系。
2. 齐次(变换)矩阵
Homogeneous Transformation Matrices

4 by 4 matrices:

Can be pre- or post-multiplied Easy to find inverse of the matrix Represents both orientation and position information, including directional vectors
Representation of a Pure Translation
1 0 T 0 0 0 1 0 0
z
a
d
0 dx 0 dy 1 dz 0 1
a o n n o
p
x
y
【纯平移变换的表示】
Representation of a Pure Translation
x
Z Z’
z
a o
p
Fobject
n
y
L a n o O Y’ Y X’
连杆的位姿可用以下齐次矩阵表示：
T n o a
nx n P y nz 0
ox oy oz 0
ax ay az 0
px py pz 1
X
P(O’)
【刚体的表示】
Frame representation Requirements
pn p sin o cos pa 0
l1
py
pxyz Rot ( x, ) pnoa
0 1 Rot (x, ) 0 cos 0 sin
sin cos 0
【绕轴纯旋转变换的表示】
z
PAB (Bx Ax )i ( By Ay ) j ( Bz Az )k
ax P b y cz
P
----【矩阵】
cz by ax
x
y
【空间向量的表示】
Application of a scale factor


Makes the matrix 4 by 1 Allows for introducing directional vectors
nx n F y nz 0
ox oy oz 0
ax ay az 0
px py pz 1
x
z
a
运动坐标系 Fn,o,a
o
p
n
y
全局参考坐标系
Fx, y , z
【坐标系在参考坐标系的表示】
Representation of a Rigid Body
nx n y nz 0 ox oy oz 0 ax ay az 0 px py pz 1
z
Axis
px pn p y l1 l2 po cos pa sin pz l3 l4 po sin pa cos
l3
pz
a

p
po pa po
pa
o
l4

y
l2
0 px 1 p 0 cos y pz 0 sin
相对于固定的参考坐标系的每次变换，变换矩阵都是左乘的。
Transformations Relative to the Rotating
(current) Frame
当进行相对于运动坐标系或当前坐标系的轴的变换时： 为计算当前坐标系中点的坐标相当于参考坐标系的变化，这时 需要右乘变换矩阵而不是左乘。
【复合变换的表示】

1. 2.
Example:
Rotation of degrees about the x-axis, Followed by a translation of [l1,l2,l3] (relative to the x-, y-, and z-axes respectively), Followed by a rotation of degrees about the y-axis.
Rotation Matrices
1 0 Rot ( x, ) 0 C 0 S 0 S C
齐次变换矩阵？
0 0 1 0 cos sin Rot x, 0 sin cos 0 0 0 0 sin cos 0 1 0 Rot y , sin 0 cos 0 0 0 cos sin 0 sin cos 0 Rot z , 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1

the three unit vectors n, o, a are mutually perpendicular each unit vector’s length, represented by its directional cosines, must be equal to 1

These constraints translate into the following six constraint equations:
A transformation may be in one of the following forms: A pure translation A pure rotation about an axis A combination of translations and/or rotations
所谓逆变换就是将被变换的坐标系返回到原来的坐标系。
o
z
Hale Waihona Puke n ao方向余弦？
x
y
全局参考坐标系 Fx , y , z
nx F n y nz