6.3.4. Coordinate Transformations in 2-D

See also

This video by Peter Corke discusses transformation matrices to apply both translation and rotation.

The calculation is different when we apply a transformation matrix to a coordinate frame to create a new, translated, and rotated coordinate frame. We read the order of transformations from left to right. To build a compound coordinate frame, we start with the previous coordinate frame and then multiply each sequential transform matrix until we reach the desired coordinate frame position.

Note

The base or world coordinate frame matrix is an identity matrix, so it is not necessary to specify it as the starting coordinate frame.

../../_images/coordTransform.png

Fig. 6.14 A transformation that traverses a straight rigid body to a new coordinate frame at the other end of the rigid body rotates the frame and then translates the frame along the x–axis.

Consider the pose at the end of a single robotic arm as shown in Fig. 6.15. Since the rotation changes the orientation of the coordinate frame, the translation is only along the \bm{x}–axis. If a rigid body also has a displacement in the direction of the \bm{y}–axis, we can either use a second translation matrix, or specify the translation matrix as having both x and y components.

{}^WT_E = \spalignmat[r]{\cos\theta, -\sin\theta, 0; \sin\theta, \cos\theta, 0; 0, 0, 1} \, \spalignmat{1, 0, a; 0, 1, 0; 0, 0, 1} = \spalignmat[r]{\cos\theta, -\sin\theta, a\,\cos\theta; \sin\theta, \cos\theta, a\,\sin\theta; 0, 0, 1} \,

../../_images/oneArm.png

Fig. 6.15 The pose of the end effector is represented by the coordinate frame \{E\}, which is defined relative to the world frame by the homogeneous coordinate transformation {}^W\mathbf{T}_E = R(\theta) Tx(a).

We read the nomenclature of the coordinate frame transformation {}^W\mathbf{T}_E as the transformation with respect to the world frame, \{W\}, yielding frame \{E\}. The reference frame is a prescript to the transformation symbol \mathbf{T}, while the label of the resulting frame is a subscript.

For serial-link robot arms, we define a coordinate frame for each joint of the robot. As shown in Fig. 6.16, the composite coordinate frames are products of the transformation matrices from the world coordinate frame to each joint.

../../_images/twoArm.png

Fig. 6.16 A coordinate frame for each joint is created by multiplicaiton of transform matrices.

If we know the position of a point relative to a transformed coordinate frame, then the point’s location in the world coordinate frame is the product of the transformation matrices and the point location in the transformed coordinate frame. However, if we know a point location in the world coordinate frame and want its location in another coordinate frame, we multiply the point location by the inverse of the transformation.

>> q1 = 30; % degrees
>> q2 = -15
>> a = 8
>> b = 3.5000
>> RA
RA =
    0.8660   -0.5000         0
    0.5000    0.8660         0
         0         0    1.0000
>> TA
TA =
    1     0     8
    0     1     0
    0     0     1
>> WTA = RA*TA
WTA =
    0.8660   -0.5000    6.9282
    0.5000    0.8660    4.0000
         0         0    1.0000
>> RB
RB =
    0.9659    0.2588         0
   -0.2588    0.9659         0
         0         0    1.0000
>> TB
TB =
    1.0000         0    3.5000
         0    1.0000         0
         0         0    1.0000
>> ATB = RB*TB
ATB =
    0.9659    0.2588    3.3807
   -0.2588    0.9659   -0.9059
         0         0    1.0000
>> B_frame = WTA*ATB
B_frame =
    0.9659   -0.2588   10.3089
    0.2588    0.9659    4.9059
         0         0    1.0000

% point in B to World
>> p1_B = [1;1;1]; % homogeneous
>> p1_W = B_frame*p1_B
p1_W =
   11.0161
    6.1306
    1.0000

% point in World to B
>> p2_W = [12;7;1];
>> p2_B = inv(B_frame)*p2_W
p2_B =
    2.1754
    1.5851
    1.0000