## 2 Mechanism Description  
During the docking process of the solar wing and satellite, the solar wing is fixed, and the position and posture of the satellite is adjusted in real time by a posture adjustment platform. The docking process can be divided into two stages according to the motion range of the satellite, that is, the macro-motion stage and the micro-motion stage. A movable heavy-load reconfigurable redundant posture adjustment platform (RrPAP) with dual motion modes is proposed in our early work. The RrPAP consists of a wheeled mobile platform (WMP) and a reconfigurable parallel posture adjustment mechanism (PAM), as shown in Figure 1. The wheeled mobile platform (WMP) is used in the macro-motion stage, while the PM is used as a posture adjustment mechanism (PAM) in the micro-motion stage. The parallel posture adjustment mechanism consists of a moving platform, a fixed platform, and four limbs connecting the two platforms, as shown in Figure 2. Each of the  
Figure 1: A movable heavy-load reconfigurable redundant posture adjustment platform (RrPAP) with dual motion modes. four limbs is a 6-DOF serial mechanism with SPPP configuration. These limbs are arranged in a rectangular layout and connected to the fixed platform above via spherical joints. \(S_{i}\) (where \(i=1\), 2, 3, 4) denotes the center of the spherical pair connecting the limb to the fixed platform. \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\) form the fixed platform of the mechanism, with \(O\) representing the geometric center of the fixed platform and \(P\) the geometric center of the adjustment platform. The upper ends of the limbs are modularly designed to connect with the moving platform. In each limb, the second prismatic pair (\(P_{2}\)) and the third prismatic pair (\(P_{3}\)) move perpendicularly to each other and parallel to the moving platform. The angles between these two prismatic pairs and the first and second primary movement directions are each \(45^{\circ}\). In the initial pose, the first prismatic pair (\(P_{1}\)) of each limb is not perpendicular to the fixed platform, forming an angle (\(\theta\)). The value of \(\theta\) is determined based on the self-locking principle. The four limbs constrain the primary motion characteristics of the mechanism, providing the moving platform with six degrees of freedom: three rotational and three translational. This parallel mechanism offers advantages such as geometric symmetry and modularity. However, the configuration of each limb with three links and four joints results in more parameters compared to chains with two links and three joints, increasing the complexity of parameter optimization. A right-handed orthogonal coordinate system is established as shown in Figure 2. A fixed coordinate system \(Oxyz\) is set up on the fixed platform, with the origin at the geometric center \(O\). The \(x\)-axis points from \(S_{4}\) to \(S_{1}\) in the initial state, the \(y\)-axis points from \(S_{1}\) to \(S_{2}\) (perpendicular to \(S_{4}S_{1}\) in the initial state), and the \(z\)-axis is determined by the right-hand rule from the \(x\) and \(y\) axes. A moving coordinate system \(Px^{\prime}y^{\prime}z^{\prime}\) is established on the moving platform, with the origin at the geometric center \(P\). The \(x^{\prime}\)-axis is perpendicular to one side of the moving platform, the \(z^{\prime}\)-axis points upwards perpendicular to the moving platform, and the \(y^{\prime}\)-axis is determined by the right-hand rule. Each limb has a limb coordinate system \(O_{\text{i}}x_{\text{i}}y_{\text{i}}z_{\text{i}}\), with the origin \(O_{\text{i}}\) at the center of the spherical pair \(S_{\text{i}}\) and the \(x_{\text{i}}y_{\text{i}}z_{\text{i}}\) directions aligned with the fixed coordinate system. In the initial state, the \(z\)-axis of the moving coordinate system coincides with that of the fixed coordinate system, while the \(x\) and \(y\) axes are parallel to their respective counterparts.