#### 3.3.3. Exponential Coordinates of Rigid-Body Motion

Any rigid-body transformation can be achieved from any other by following some 6-vector twist for unit time. The six coordinates of this twist are called the exponential coordinates. This video shows how the rigid-body transformation can be calculated using a matrix exponential with the se(3) matrix representation of the exponential coordinates. The matrix exponential maps an element of se(3) (the matrix representation of a twist) to an element of SE(3) (the 4×4 transformation matrix representing a rigid-body configuration) and the matrix logarithm maps an element of SE(3) to the element of se(3) that achieves it when integrated for unit time.

In the previous videos, we learned that any instantaneous velocity of a rigid body can be represented as a twist, defined by a speed theta-dot rotating about, or translating along, a screw axis S. In this video, we integrate the vector differential equation describing the motion of a frame twisting along a constant screw axis to find the final displacement of the frame.

This animation shows a screw axis S and a frame at time zero. Let's say that this configuration is coincident with the space frame {s}, so its representation is the identity matrix. Now we let the frame twist about the screw axis at a rotational speed theta-dot = 1. The animation shows the configuration achieved at times t equals pi and t equals 2 pi. These configurations are represented as transformation matrices, but an alternative representation would use 6 exponential coordinates, similar to the 3 exponential coordinates for rotation that we saw earlier. In this case, we could represent the configuration at time pi as S times pi, meaning the configuration achieved after the frame has followed the screw axis S for time pi. Similarly, we could represent the configuration at time 2 pi as S times 2 pi.

Let's look at some of the analogies between rotations and full rigid-body motions. For rotations, we have a unit rotation axis omega-hat. For rigid-body motions, we have a screw axis where either the angular component is a unit vector or the angular component is zero and the linear component is a unit vector.

For rotations, the exponential coordinates are omega-hat theta, where theta is the angle of rotation about the axis omega-hat. For rigid-body motions, the exponential coordinates are S-theta. If the screw axis has any angular component, theta is the angle rotated about the screw axis. If the screw axis has zero rotation, then theta is the linear distance traveled along the axis.

For rotations, the matrix representation of the exponential coordinates is the 3 by 3 skew symmetric representation of omega-hat times theta. The set of all such matrices is called little so(3). For rigid-body motions, the matrix representation of the exponential coordinates is a 4 by 4 matrix in little se(3), which we learned about in the last video.

For rotations, the exponential maps matrices in little so(3) to rotation matrices, and the log maps rotation matrices to little so(3). For rigid-body motions, the exponential maps matrices in little se(3) to transformation matrices, and the log maps transformation matrices to little se(3).

As with the case for rotations, the matrix exponential for rigid-body motions has closed-form solutions. There are two cases to consider: one where the screw axis is a pure translation with no rotation, and one where the screw axis has rotation. For the case of a purely translational screw axis, theta refers to the linear distance traveled, and the solution is particularly simple: the orientation is unchanged, hence the identity matrix in the top left submatrix, and the new position is just the unit linear velocity of the screw axis times the distance traveled. For the case of a screw axis with rotation, again a closed-form solution exists, but it is a bit more complicated.

The algorithm for the matrix logarithm involves inverting these expressions to find the matrix representation of the exponential coordinates S times theta.

Now, given a body frame {b} at the configuration T_sb relative to the space frame {s}, we would like to know the final configuration T_sb-prime of the body frame if it travels a distance theta along the screw S. We could represent S in either the {s} frame or the {b} frame. If we define it in the {b} frame, the final configuration T_sb prime is T_sb times the matrix exponential. Remembering that multiplying by a transformation on the right corresponds to a transformation expressed in the frame of the second subscript. If we define the screw axis in the {s} frame, we must premultiply T_sb by the matrix exponential, since multiplication on the left means that the transformation is expressed in the frame of the first subscript.

Each single-degree-of-freedom joint of a robot, such as a revolute joint, a prismatic joint, or a helical joint, has a joint axis defined by a screw axis. The matrix exponential and log will be used extensively in the study of robot kinematics, starting in Chapter 4.

The next and final video of Chapter 3 covers the representation of forces and torques in three-dimensional space.