# Design and Motion Planning of a Biped Climbing Robot with Redundant Manipulator
## 4 Experiments  
To verify the effectiveness of the proposed robotic system and related motion planning algorithms, a series of simulations and experiments with the CMBOT were conducted. As shown in Figure 14, a robot motion simulation system for the CMBOT was developed to examine the robot trajectories for obvious problems before the trajectories are transferred to the actual robot system, thus ensuring the safety of the robot motion. The motion planning algorithms of the climbing mechanism and manipulator were simulated using the proposed simulation software. When planning the motion of the climbing mechanism, the following foot trajectory parameters were selected: \(S=240\) mm, \(H=50\) mm, \(D=0\), \(T_{\mathrm{b}}=10\) s, \(\varphi_{\mathrm{y}}=0\), \(\varphi_{\mathrm{z}}=0\). According to Equation (8),  
Figure 14: Snapshot of the motion simulation system for the CMBOT. the curves of displacement, velocity, and acceleration of the low-contact impact trajectory during the extension phase are shown in Figure 15. As shown in Figure 15, the proposed trajectory is smooth so there is not a sudden change in the velocity or acceleration of the trajectory, and both the velocity and acceleration will be equal to 0 when the magnetic feet contact the handrail. This is very important for reducing the impact when the magnetic feet contact the handrail. The joint angles of the climbing mechanism during the motion are shown in Figure 16.  
The proposed motion planning algorithm of the redundant manipulator was also verified in the simulation. One of the most commonly used trajectories will be shown in the simulation. The end of the manipulator will track this selected trajectory from the start point to the side of the bracket. The position and pose angle of the end of the manipulator at the start point is (0 mm, 820 mm, -2790 mm, 0\({}^{\circ}\), 0\({}^{\circ}\), -90\({}^{\circ}\))T in coordinate (6). The simulation will last 20 s, and equations of the desired trajectory can be calculated by:  
\[\dot{q_{\rm x}}=\ \left\{ \begin{array}{c}80t-450,\ \ 0\leq t<5\\ 0,\ \ 5\leq t<20\end{array} \right. \tag{21}\]  
\[\dot{q_{\rm y}}=\ \left\{ \begin{array}{c}-160,\ \ 0\leq t<5\\ -78,\ 5\leq t<20\end{array} \right. \tag{22}\]  
\[\dot{q_{\rm z}}=\ \left\{ \begin{array}{c}98t,\ \ 0\leq t<5\\ 0,\ 5\leq t<20\end{array} \right. \tag{23}\]  
\[\omega_{\rm x}=\ \left\{ \begin{array}{c}7.34t-18.35,\ \ 0\leq t<5\\ 0,\ \ 5\leq t<20\end{array} \right. \tag{24}\]  
\[\omega_{\rm y}=\ \left\{ \begin{array}{c}28.8t,\ \ 0\leq t<5\\ 0,\ 5\leq t<20\end{array} \right. \tag{25}\]  
Figure 16: The joint angles of the climbing mechanism. Figure 15: Curves of the low-contact impact trajectory: (**a**) Displacement, (**b**) Velocity, and (**c**) Acceleration. According to Equations (21)-(26), the curves of position and pose angle of the end of the manipulator in the simulation are shown in Figures 17 and 18. In the first 5 s of the simulation, the end of the manipulator move forms the start point to working position along the spatial curve, and then move along a straight line at the root of the bracket. The information of the environment should be acquired before we calculate the joint angles of the redundant by the proposed motion planning algorithm. Firstly, with the help of the long manipulator, the LIDAR sensor on the manipulator can scan the working surfaces from some certain positions where it is far enough from the obstacles of the railway bridge. The 3D point data will be transformed into coordinate (6) (i.e., the base of the manipulator) according to the manipulator configuration. Then 3D point cloud from different visual angles will be combined to obtain more complete information about the bridge. The combined 3D point data needs to be filtered to remove clutter and make the data be distributed evenly before further processing. Figure 19 shows the processed 3D point cloud of the railway bridge.