# Multitask-Based Trajectory Planning for Redundant Space Robotics Using Improved Genetic Algorithm
## 2 Preliminaries and Notation  
### Kinematic Models of Space Robotic Systems  
The 2D representation of a space robotic system is outlined in Figure 1, where the system consists of \(n+1\) bodies \(B_{i}(i=0,...,n)\). The coordinate frame \(\Sigma_{I}\) coincides with the coordinate frame \(\Sigma_{0}\) at initial time, and the coordinate frame \(\Sigma_{i}\) (\(i=1,...,n\)) is established using the D-H approach [33; 34]. The D-H approach aims at establishing a coordinate frame fixed on each link and computing the coordinate transformation between adjacent coordinate frames by four D-H parameters. The coordinate frame \(\Sigma_{i}\) (\(i=1,...,n\)) is defined as follows: \(Z_{i}\)-axis is chosen along rotational direction of \((i+1)\)th joint, origin \(O_{i}\) is located at intersection of \(Z_{i}\)-axis with common perpendicular line of \(Z_{i-1}\)- and \(Z_{i}\)-axis, \(X_{i}\)-axis is chosen along common perpendicular line of \(Z_{i-1}\)- and \(Z_{i}\)-axis with positive direction from \(i\)th joint to \((i+1)\)th joint, and \(Y_{i}\)-axis is chosen to follow the right-handed rule. After the establishment of \(\Sigma_{i}\) (\(i=1,...,n\)), four D-H parameters \(a_{i}\), \(\alpha_{i}\), \(d_{i}\), and \(\theta_{i}\) are specified as follows: \(a_{i}\) denotes the distance between \(Z_{i}\)- and \(Z_{i+1}\)-axis along \(X_{i}\)-axis; \(\alpha_{i}\) denotes the angle from \(Z_{i}\)- to \(Z_{i+1}\)-axis around \(X_{i}\)-axis; \(d_{i}\) denotes the distance between \(X_{i-1}\)- and \(X_{i}\)-axis along \(Z_{i}\)-axis; and \(\theta_{i}\) denotes the angle from \(X_{i-1}\)- to \(X_{i}\)-axis around \(Z_{i}\)-axis. Two categories of space robotic systems are usually employed for manipulation tasks, the FFSR1 and the FFSR2, which are collectively called FFSR in Figure 1. The base attitude of FFSR1 is controlled, while the base of FFSR2 is in free-floating mode. If both the position and attitude of the base are fixed, the differential kinematic equation is formulated as:  
\[\left[ \begin{array}{c}\mathbf{v}_{e}\\ \mathbf{\omega}_{e}\end{array} \right]=\mathbf{J}_{m}\mathbf{\dot{q}} \tag{1}\]  
\(J_{m}\) in (1) is identical to the Jacobian matrix of industrial robotics. FFSR1 satisfies the linear momentum-conservation law, contributing to the following differential kinematic equation:  
\[\left[ \begin{array}{c}\mathbf{v}_{e}\\ \mathbf{\omega}_{e}\end{array} \right]=\left[ \begin{array}{c}-\frac{1}{M}I_{bb}J _{iw}+J_{mv}\\ J_{mv}\end{array} \right]\mathbf{\dot{q}} \tag{2}\]  
However, FFSR2 satisfies the linear and angular momentum-conservation laws shown in Equation (3), where momentum value is usually set to 0 for simplicity. \[\mathbf{H}_{b}\mathbf{\dot{x}}_{b}+\mathbf{H}_{bm}\mathbf{\dot{q}}=\mathbf{0} \tag{3}\]  
Figure 1: Two-dimensional sketch of a space robotic system. Since matrix \(\mathbf{H}_{b}\) in Equation (3) is symmetric and positively definite, the velocity of the free-floating base can be expressed as:  
\[\dot{\mathbf{x}}_{b}=\left[ \begin{array}{c}\mathbf{v}_{b}\\ \mathbf{\omega}_{b}\end{array} \right]=-\mathbf{H}_{b}^{-1}\mathbf{H}_{bm}\mathbf{\dot{q}}=\mathbf{J} _{bm}\mathbf{\dot{q}} \tag{4}\]  
Therefore, the differential kinematic equation of FFSR2 can be expressed as:  
\[\left[ \begin{array}{c}\mathbf{v}_{e}\\ \mathbf{\omega}_{e}\end{array} \right]=\mathbf{J}_{b}\left[ \begin{array}{c}\mathbf{v}_{b }\\ \mathbf{\omega}_{b}\end{array}