In this section, some fundamental geometric concepts pertaining to the formulation of the calibration model are presented.

Transformation Matrices and Twist Representation

The special Euclidean group, denoted by SE(3) ⊂ R4x4, of rigid body motions consists of matrices of the general form

where R belongs to the special orthogonal group, denoted by SO(3) ⊂ ℝ3x3, and describes a rigid body rotation and t = [tx, ty, tz]T ⊂ ℝ3 describes a rigid body translation.
The Lie algebra of SE(3), denoted by se(3) ⊂ ℝ4x4, consists of matrices with the following form

where ω ∈ so(3) ⊂ ℝ3x3 is a skew symmetric matrix with

is an element in the Lie algebra of SO(3), denoted by so(3), and v ∈ ℝ3. Note that an element 𝜔̂ ∈ so(3) can also be represented in the corresponding vector form ω given by

with 𝜔̂ defined in Eq. 3. An element ŷ ∈ se(3) can thus be represented in corresponding vector form y given by

termed a twist. The twist represents the line coordinate of the screw axis of a general rigid body motion, in which ω and v are the direction and position vectors of the screw axis, respectively.

Adjoint Representation for Lie Group and Lie Algebra

An element of a Lie group can also be identified with a linear mapping between its Lie algebra via the adjoint representation. For an element in the Lie group, X ∈ SE(3) defined in Eq. 1, its adjoint map acting on an element in its Lie algebra,  ŷ ∈ se(3) defined in Eq. 2, is given by

If y is the vector representation of ŷ, then the adjoint map of X ∈ SE(3) acting on an element y is given by

where t̂  is given by

and AdX(y) is the vector representation of AdX(ŷ).

Matrix Exponentials

An important connection between a Lie group, SE(3), and its Lie algebra, se(3), is the exponential mapping, defined on each Lie algebra. A general definition of matrix exponential for ŷ ∈ ℝ4x4 is given by

where Inxn ∈ ℝnxn represents an identity matrix of dimension n.
If ŷ ∈ se(3) as defined in Eq. 2, an explicit formula for the exponential mapping of ŷ is given by (Okamura and Park 1996), then

where ||•|| denotes the Euclidean norm of  •.

Product of Exponentials

In this chapter, the product-of-exponentials (POE) formula (Brockett 1984; Murray et al. 1994; Park and Brockett 1994) is adopted for expressing the forward kinematics of an open chain robot work cell containing either revolute or prismatic joints where the forward kinematics equation is expressed as a product of matrix exponentials.
The ith frame, denoted by {Ki}, is attached to the ith link and the consecutive (i - 1)th and ith links are connected by the ith joint. The relative pose of the ith frame with respect to the (i - 1)th frame under a joint displacement qi can be described by a 4 x 4 homogeneous matrix, an element of SE(3), as follows:

where i-1Ti(0) ∈ SE(3) denotes the relative pose of the frame {Ki} with respect to the frame {Ki-1} for qi = 0, ŝi ∈ se(3) is the twist associated with joint i and when expressed in {Ki} is given by

The twist ŝi can be expressed as a vector through a mapping ŝi → si = [viT ,ωiT]T ∈ ℝ6 with ωi = [ωix, ωiy, ωiz]T ∈ R3 and si is a unit vector defined as the twist coordinates of joint i expressed in {Ki}. The definition of eŝi ∈ SE(3), the exponential of ŝi ∈ se(3), is given in section “Matrix Exponentials.” In this chapter, the symbol Π is used to represent a series of noncommutative products of matrices where

For a robot with n joints, as illustrated in Fig. 2 for the case n = 6, the transformation describing the relative pose of the sensor frame, {Kn+1}, in the measurement frame, {K1}, is given by

where q = [q1, . . ., qn]T. There exists an element p̂i ∈se(3) such that ep̂i = i-1Ti(0) ∈ SE(3).
Note that the element pi ∈ R6 is a vector which will be fine-tuned in the calibration process to enhance the accuracy of the robot kinematic model. As such, 0Tn+1(q) in Eq. 11 can also be expressed in the following form:

More information on matrix exponentials and the product-of-exponential approach can be obtained from (Brockett 1984; Murray et al. 1994; Park and Brockett 1994).