# Multitask-Based Trajectory Planning for Redundant Space Robotics Using Improved Genetic Algorithm
## 2 Preliminaries and Notation
### Minimum Base Attitude Disturbance
#### 2.2.2 Depiction of Joint Configurations Corresponding to Each Waypoint  
The inverse kinematics of redundant space robotics are complex, since one point in the task space corresponds to infinite points in the joint space. An approach based on the geometric construction of the robot, especially the robot called Spherical Revolute Spherical (SRS) [32; 35; 36; 37], is well developed. Each component of the joint configuration is formulated as a function of the arm angle. By assigning a value to the arm angle, a finite number of joint configurations are obtained. Figure 2 presents the process for constructing the arm angle. The following legend is useful to understand the different subfigures. Figure 2a outlines the 7-DOF rotational robot SRS based on the D-H approach, together with the three main points \(S\), \(E\), and \(W\). The shoulder point \(S\) denotes the intersection point of \(Z_{i}\)-axis (\(i=0,1,2\)), the elbow point \(E\) denotes the origin \(O_{3}\) of \(\Sigma_{3}\), and the wrist point \(W\) denotes the intersection point of \(Z_{i}\)-axis (\(i=4,5,6\)). Figure 2b outlines the reference plane constructed by \(Z_{0}\)-axis and \(\omega\), the arm plane \(SEW\), and the arm angle \(\psi\) from the reference plane to the arm plane. Figure 2c coincides with Figure 2b, where \(\omega\) denotes the vector from \(S\) to \(W\). \(k\) and \(e\) denotes the vectors perpendicular to \(\omega\) in the reference plane and the arm plane, respectively. Figure 2d outlines a case \(\psi=0\), meaning that the reference plane coincides with the arm plane. The following procedure is carried out for the derivation of the joint configuration \(\mathbf{q}=\left[q_{1},...,q_{7}\right]^{T}\). **Step 1**: Construction of the arm plane, the reference plane and the arm angle. First, the arm plane is constructed according to the points \(S\), \(E\), and \(W\) shown in Figure 2. Second, the reference plane is constructed based on \(\omega\) and \(Z_{0}\)- or \(X_{0}\)-axis, since at least one of \(Z_{0}\)- and \(X_{0}\)-axes is not coincident with \(\omega\). The arm angle \(\psi\) is then depicted, shown in Figure 2.
**Step 2**: Derivation of the joint angle \(q_{4}\). Project the arm plane \(SEW\) to the plane perpendicular to the rotation axis of the elbow joint, namely \(Z_{3}\)-axis. According to the geometric construction of SRS and the Pythagorean theorem, two values of \(q_{4}\) are derived. * Derivation of the shoulder-joint angles \(q_{1}\), \(q_{2}\), and \(q_{3}\). First, the rotation transformation, rotating around \(\omega\) with the rotation angle \(\psi\), is derived. Second, the orientation of \(\Sigma_{3}\) relative to \(\Sigma_{0}\) using the concept of the shoulder reference attitude matrix [32]. Finally, two groups of shoulder-joint angles are derived. * Derivation of the wrist-joint angles \(q_{5}\), \(q_{6}\), and \(q_{7}\). Two groups of wrist-joint angles are derived according to the predefined attitude of the end effector, the rotation transformation and the should reference attitude matrix derived in **Step 3**, and the rotation matrix from \(\Sigma_{3}\) to \(\Sigma_{4}\). **Remark 2**. : _A detailed explanation about the approach of computing \(\mathbf{q}\) is expressed in [32]. According to the aforementioned procedure, eight groups of \(\mathbf{q}\) are derived with an assignment of the arm angle \(\psi\). \(\psi\) is set to zero considering the following aspects. First, the calculation accuracy of \(\mathbf{q}\) corresponding to each waypoint declines as the absolute value of \(\psi\) increases. The result of \(\mathbf{q}\) is derived with the highest accuracy in the case \(\psi=0\). Second, the assignment of \(\psi=0\) contributes to less computational cost [32]. Besides, the angular velocity of each joint is constrained to be zero at each waypoint, and is not affected by values of \(\psi\)._  
**Remark 3**.