# Sampling-Based Motion Planning for Free-Floating Space Robot without Inverse Kinematics
## 3 Motion Planning Algorithm for FFSRs
The local planner based on the general Jacobian matrix chooses actions according to the EE pose error in each iteration:

\[\dot{\mathbf{\Theta}}\ =\ \Big{(}\boldsymbol{J}_{g}\Big{)}^{+}\Delta x_{e} \tag{25}\]

The local planner based on the extended Jacobian matrix chooses actions according to the EE pose error and the base attitude error in each iteration:

\[\dot{\mathbf{\Theta}}\ =\ \Bigg{(}\ \begin{bmatrix}\boldsymbol{H}_{\omega}\\ \boldsymbol{J}_{g}\end{bmatrix} \Bigg{)}^{+}\Bigg{[}\ \begin{matrix}\Delta\boldsymbol{\Psi}_{b}\\ \Delta x_{e}\end{matrix} \Bigg{]} \tag{26}\]

For these three local planners, the configuration parameters at the initial time are all set to \(0^{\lx@math@degree}\). The centroid position of the system is set to \(\Big{[} \begin{array}{ccc}0&0&0\end{array} \Big{]}^{T}m\). The base attitude disturbance range is set as \(\pm 35^{\circ}\). The joint angle limit is set as \(\pm 300^{\circ}\). The reference goal base attitude is set as \(\Big{[} \begin{array}{ccc}0&0&0\end{array} \Big{]}^{\circ}\) for the local planner for goal EE pose guiding growth proposed in this paper. For these three local planners, the goal EE position is \(\Big{[} \begin{array}{ccc}6.8&1&2\end{array} \Big{]}^{T}m\) and the goal EE attitude is \(\Big{[} \begin{array}{ccc}50&60&70\end{array} \Big{]}^{\circ}\). Simulation results are shown in Figures 1-3. From these results, we can see that the local planner based on the general Jacobian matrix can well achieve the EE pose goal, but the attitude of the base cannot meet the limit for this local planner. The local planner based on the extended Jacobian matrix can adjust the base attitude to a very low value. As the extended Jacobian matrix has more rows than columns, the local planner based on the extended Jacobian matrix cannot eliminate the EE pose error. Actually, for an FFSR, we only need to ensure that the base attitude disturbance meets the requirements. The main task for the motion planning of FFSRs is to drive its EE to a goal pose; there is no need to adjust its base attitude all the time. The local planner for the goal EE pose guiding growth proposed in this paper can not only ensure that the base attitude disturbance meets the requirements but also drive EE to a goal pose. Figure 1: Time histories of EE position error, EE attitude error, base attitude trajectory, and joint trajectory for the local planner based on the general Jacobian matrix. (**a**) Time histories of EE position error; (**b**) time histories of EE attitude error; (**c**) time histories of base attitude trajectory; (**d**) time histories of joint trajectory. Figure 2: Time histories of EE position error, EE attitude error, base attitude trajectory, and joint trajectory for local planner based on the extended Jacobian matrix. (**a**) Time histories of EE position error; (**b**) time histories of EE attitude error; (**c**) time histories of base attitude trajectory; (**d**) time histories of joint trajectory. ### Verification of the Effectiveness of the Proposed RRTs for Free-Floating Space Robots

The simulation for the second part is divided into two scenarios: (1) Starting from the initial configuration and finally reaching the target EE position; (2) starting from the initial configuration and reaching the target EE pose. For these two scenarios, the configuration parameters at the initial time are all set to \(0^{\lx@math@degree}\). The centroid position of the system is set to \(\left[ \begin{array}{cccc}0&0&0\\ \end{array} \right]^{T}\!m\). The base attitude disturbance range is set as \(\pm 35^{\circ}\). The joint angle limit is set as \(\pm 300^{\lx@math@degree}\). The reference goal base attitude is set as \(\left[ \begin{array}{cccc}0&0&0\\ \end{array} \right]^{\lx@math@degree}\). For Scenario 1, the goal EE position is \(\left[ \begin{array}{cccc}4&2&1\\ \end{array} \right]^{T}\!m\). For Scenario 2, the goal EE position is \(\left[ \begin{array}{cccc}4&3&1\\ \end{array} \right]^{T}\!m\), and the goal EE attitude is \(\left[ \begin{array}{cccc}50&60&70\\ \end{array}