The product of exponentials (POE) method is a robotics convention for mapping the links of a spatial kinematic chain. It is an alternative to Denavit–Hartenberg parameterization. While the latter method uses the minimal number of parameters to represent joint motions, the former method has a number of advantages: uniform treatment of prismatic and revolute joints, definition of only two reference frames, and an easy geometric interpretation from the use of screw axes for each joint. The POE method was introduced by Roger W. Brockett in 1984.
Method The following method is used to determine the product of exponentials for a kinematic chain, with the goal of parameterizing an affine transformation matrix between the base and tool frames in terms of the joint angles θ 1 . . . θ N . {\textstyle \theta _{1}...\theta _{N}.}
Define "zero configuration" The first step is to select a "zero configuration" where all the joint angles are defined as being zero. The 4x4 matrix g s t ( 0 ) {\textstyle g_{st}(0)} describes the transformation from the base frame to the tool frame in this configuration. It is an affine transform consisting of the 3x3 rotation matrix R and the 1x3 translation vector p. The matrix is augmented to create a 4x4 square matrix.
g s t ( 0 ) = [ R p 0 1 ] {\displaystyle g_{st}(0)=\left[{\begin{array}{cc}R&p\\0&1\\\end{array}}\right]}
Calculate matrix exponential for each joint The following steps should be followed for each of N joints to produce an affine transform for each.
Define the origin and axis of action For each joint of the kinematic chain, an origin point q and an axis of action are selected for the zero configuration, using the coordinate frame of the base. In the case of a prismatic joint, the axis of action v is the vector along which the joint extends; in the case of a revolute joint, the axis of action ω the vector normal to the rotation.
Find twist for each joint A 1x6 twist vector is composed to describe the movement of each joint. For a revolute joint, ξ i = ( ω i − ω i × q i ) . {\displaystyle \xi _{i}=\left({\begin{array}{c}\omega _{i}\\-\omega _{i}\times q_{i}\\\end{array}}\right).}
For a prismatic joint, ξ i = ( 0 v i ) . {\displaystyle \xi _{i}=\left({\begin{array}{c}0\\v_{i}\\\end{array}}\right).}
The resulting twist has two 1x3 vector components: Linear motion along an axis ( v {\displaystyle v} ) and rotational motion along the same axis (ω). ξ = ( ω v ) . {\displaystyle \xi =\left({\begin{array}{c}\omega \\v\\\end{array}}\right).}
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