We can represent the configuration of a body frame {b} in the fixed space frame {s} by specifying the position p of the frame {b}, in {s} coordinates, and the rotation matrix R specifying the orientation of {b}, also in {s} coordinates. We gather these together in a single 4 by 4 matrix T, called a homogeneous transformation matrix, or just a transformation matrix for short. The bottom row, which consists of three zeros and a one, is included to simplify matrix operations, as we'll see soon.

The set of all transformation matrices is called the special Euclidean group SE(3). Transformation matrices satisfy properties analogous to those for rotation matrices. Each transformation matrix has an inverse such that T times its inverse is the 4 by 4 identity matrix. The product of two transformation matrices is also a transformation matrix. Matrix multiplication is associative, but not generally commutative.

Also analogous to rotation matrices, transformation matrices have three common uses: The first is to represent a rigid-body configuration. The second is to change the frame of reference of a vector or a frame. The third is to displace a vector or a frame.

To represent a frame {b} relative to a frame {s}, we construct the matrix T_sb consisting of the rotation matrix R_sb, as we saw in previous videos, and the position p of the {b} frame origin in {s} frame coordinates. The representation of the {s} frame relative to the {b} frame is just the inverse. As with the rotation matrix, the matrix inverse corresponds to switching the order of the subscripts.

To change the frame of reference of a configuration, we can use the same subscript cancellation rule as for rotation matrices. If we know T_sb and T_bc, we can calculate T_sc, representing the configuration of frame {c} in frame {s}, by multiplying T_sb by T_bc. The inverse of T_sc is T_cs. Just as we followed T_sb and then T_bc to get to T_sc, we can follow Tbc inverse and T_sb inverse to get T_cs.

We can also change the frame of reference for a point p in space. Let p_b and p_s be the representations of the point in the {b} and {s} frames. We could naively try our subscript cancellation rule again, but this doesn't work: T_sb and p_b have a dimension mismatch. To fix this, we simply append a 1 to the end of each vector, making the 3-vector into a 4-vector. This is called the homogeneous coordinate representation of the 3-vector.

Finally, a transformation matrix can be used to displace a point or a frame. Consider the fact that any configuration can be achieved from the initial configuration by first rotating, and then translating. In this animation, a frame initially at the zero orientation rotates about a fixed axis omega-hat a distance theta. It then translates according to the vector p, which is expressed in the coordinates of the initial frame T_zero. Its final configuration is given by T, where the Translation and Rotation operators are expressed by these matrices. T can be viewed not only as a configuration, but also as the transformation that takes the identity matrix to T.

Let's consider a specific example of using a transformation matrix T to move a frame. Our transformation T is defined by a translation of 2 units along the y-axis, a rotation axis aligned with the z-axis, and a rotation angle of 90 degrees, or pi over 2. We will use the transformation T to move the {b} frame relative to the {s} frame. The {b} frame is initially represented by T_sb.

Since we have two frames, we need to know whether the transformation vectors p and omega-hat are expressed in the {b} frame or the {s} frame. The answer depends on whether T right-multiplies or left-multiplies T_sb.

If we left-multiply T_sb by T, the vectors p and omega-hat are considered to be expressed in the frame of the first subscript of T_sb, the {s} frame. Let's animate the transformation T. The rotation axis z and the translation axis y, expressed in the {s} frame, are shown. First the {b} frame will rotate 90 degrees about the z-axis of the {s} frame, and then it will translate 2 units along the y-direction of the {s} frame. Let's run the animation. And now one more time. Notice where the {b} frame ends up. We call this new frame {b-prime}.

If instead we right-multiply T_sb by T, the vectors p and omega-hat are considered to be expressed in the frame of the second subscript of T_sb, the {b} frame. Also, the order of the operations is reversed: first we translate T_sb, and then we rotate it. Let's animate the motion. Watch how the {b} frame first translates by 2 units in the y-direction of the {b} frame, then rotates about the z-axis of the {b} frame. Let's run the animation. Notice that the body z-axis, used for rotation in the second step, moved along with the frame during the initial translation. And now one more time. Notice where the {b} frame ends up. We call this new frame {b-double-prime}.

In summary, if the transformation T is applied on the right, the vectors p and omega-hat are considered to be expressed in the body frame, moving the frame {b} to the new frame {b-double-prime}. If the transformation T is applied on the left, p and omega-hat are considered to be expressed in the space frame, moving the frame {b} to the new frame {b-prime}.

In the next video we introduce our representation of a rigid-body linear and angular velocity, called a twist.