## 4 Parameter-Identification-Based Configuration Design
### Parametric Error Compensation Model  
From the discussion above, we know that it is unfeasible to identify the precise position of the fixed point \(P\) directly. As a result, we therefore begin with a special position of \(P\) and gradually approach the expected solution, adopting the idea of parameter identification in robotics [58; 59]. Parameter identification is the process of minimizing the residuals between observed data and corresponding parametric error compensation model (PECM). In order to exploit the idea of parameter identification, the geometric model in Figure 7, which completely fulfills all of the GCs, is regarded as the observed data. To fit the observed data, a PECM is established first. We begin with a simple example in which the fixed point \(P\) is located at a special position, as shown in Figure 8a. In the deployed state, \(P\) coincides with \(D_{1}\), points \(A_{1}\) and \(D_{1}\) are precisely located on the envelope after folding and the focal axis is coplanar with points \(A_{1}\) and \(B_{1}\), suggesting that GCs 1 and 2 as well as the weak GC 4 are satisfied. However, point \(C_{1}\) fails to obtain an upward displacement \(h\); instead, it has a downward motion and interferes with the central hub. To compensate for this error, the position of \(P\) is relaxed by moving a certain distance in the radial direction on the plane of the central aperture. Meanwhile, a parameter \(k_{1}\) is introduced here to quantify the compensating effect, as shown in Figure 8b. In this way, it is conceivable that point \(C_{1}\) will have an upward motion, and that a larger \(k_{1}\) implies that the upward motion is more noticeable.  
Unfortunately, the above operation will inevitably introduce another error, where point \(D_{1}\) is not exactly located on the envelope; instead, it will be completely inside the envelope. This panel pose differentiates from that described in Figure 6c, which eventually brings about a descent in RSE. To additionally compensate for this error, another compensating strategy is introduced here. As shown in Figure 8c, the compensation parameter \(k_{2}\) is defined as \(k_{2}=\angle POD_{1}/\angle C_{1}OD_{1}\), and, by definition, the location of the fixed point \(P\) is equivalent to traveling a certain path relative to point \(D_{1}\) anticlockwise along arc \(C_{1}D_{1}\), as shown by the bottom subfigure in Figure 8c. After folding, point \(D_{1}\) will possess an outward motion and even pass through the envelope to be completely outside it, as shown in the top subfigure of Figure 8c. Except for \(k_{1}\) and \(k_{2}\), parameter \(\beta\) in Figure 7a also affects the pose of the panel, and GC 4 can be satisfied if assigning it an appropriate value.  
Given the compensating effect of parameters \(k_{1}\), \(k_{2}\) and \(\beta\), they are collectively referred to as the compensation parameters. According to the discussion above, these compensation parameters are capable of characterizing the deviation of the nominal values and observed values, thereby verifying the completeness of the PECM. Furthermore, small changes in the panel pose must correspond to small changes in the compensation parameters, by which the continuity of the model can be validated. Note that, \(k_{1}\), \(k_{2}\) and \(\beta\) are independent of each other, and thus the PECM has a minimal number of parameters; namely, it possesses the property of minimality. In summary, the PECM exhibits three distinguishing properties of completeness, continuity and minimality, thereby satisfying the basic requirements for parameter identification [60,61] (see Appendix A for a detailed explanation on the PECM properties of completeness, continuity and minimality). Therefore, all of the above four GCs will be satisfied if proper values are assigned to those compensation parameters.