## 4 The Improved Self-Collision Detection Technology
### Determine the Position of Each Approximation Object
#### 4.2.3 Determine the Position of the Mobile Platform and the Support Frame  
For the mobile platform or the support frame of the manipulator subsystem, we can directly describe their positioning by the mobile-fixed coordinate frame and the basic structural parameters, and we can also confirm the radius of the spherical approximation based on the basic parameters of the mobile platform or the support frame. For example, for the mobile platform, the positioning of the midpoint of the driving wheels of the mobile platform is expressed as \([p_{xd},p_{yd},p_{zd}]\); the positioning of the center of gravity of the driving wheel of the mobile platform is expressed as \([p_{xc},p_{yc},p_{zc}]\). Therefore, the unit vectors along the \(x\)- and \(y\)-axes of the body-fixed frame can be expressed as  
\[\begin{cases}e_{xdc}=\frac{[p_{xd},p_{yd},0]^{T}-[p_{xc},p_{yc},0]^{T}}{\|\,[p _{xd},p_{yd},0]^{T}-[p_{xc},p_{yc},0]^{T}\,\|\,};\\ e_{ydc}=e_{xdc}\times e_{z};\end{cases} \tag{15}\]  
where \(e_{xdc}\) represents the unit vector along the \(x\)-axis of the body-fixed frame; \(e_{z}=[0,0,1]\) represents the unit vector along the \(z\)-axis of the body-fixed frame; and \(e_{ydc}\) represents the unit vector along the \(y\)-axis of the body-fixed frame.  
We determined the center of each spherical object. The length, width, and height of the mobile platform are three fixed known quantities, and the radius of the corresponding myopic sphere is expressed as \(R_{s}^{M}\). Thus, the center position of multiple spherical objects can be expressed as  
\[\begin{cases}C_{ij}^{M}=C^{M}-p\cdot e_{ydc},(p=-\frac{\dot{j}}{2}:1:\frac{ \dot{j}}{2};)\\ C^{M}=[P_{xc},P_{yc},P_{zc}]^{T}-k\cdot e_{xdc},(k=-\frac{i}{2}:1:\frac{i}{2}) ;\\ i=\frac{L_{M_{i}}}{\sqrt{2}R_{s}^{M}};\dot{j}=\frac{B_{M_{i}}}{\sqrt{2}R_{s}^{ M}};\end{cases} \tag{16}\]  
where \(C_{ij}^{M}\) represents the position of the center of the corresponding approximation for the mobile platform, \(L_{M_{i}}\) represents the length of the mobile platform, and \(B_{M_{i}}\) indicates the width of the mobile platform.  
We determined the location of the surface point. After determining the center and the radius of each sphere based on the basic structure of the corresponding component and Formula (16), the location of the surface point can be acquired via the surface expression of each sphere, which can be denoted as  
\[\begin{cases}p_{xij}^{M}=C_{x}^{M}(i)+R_{s}^{M}(i)\cdot cos(m^{M});\\ p_{yij}^{M}=C_{y}^{M}(i)+R_{s}^{M}(i)\cdot sin(m^{M})\cdot cos(n^{M});\\ p_{zij}^{M}=C_{z}^{M}(i)+R_{s}^{M}(i)\cdot sin(m^{M})\cdot sin(n^{M});\end{cases} \tag{17}\]  
where \(p_{xij}^{I},p_{yij}^{I},p_{zij}^{J}\) represent the positioning result of the corresponding approximation of the mobile platform; \(C_{x}^{M}(i),C_{y}^{M}(i),{}^{M}C_{z}(i)\) represent the center of the spherical object approximated the corresponding link; and \(m^{M},n^{M}\) represent the minimum divided distance of the corresponding spherical object. The value range is \([0,2\pi],[0,\pi]\).  
After completing the positioning of each component in the three-dimensional space, the pre-processing of self-collision detection of the manipulator robot is complete. Next, we need to determine whether there is a collision between the different components.