# Model-Based Adaptive Collaboration of Multi-Terminal Internal Force Tracking
## 2 Model Establishment of a Multi-Terminal Clamping System  
### Formalizing the Workspace  
The following symbols represent the position, velocity, and force of each system part used throughout the paper. Note that in the symbols below, the superscripts and subscripts are omitted. \(h\)Vector composed of \(F\) and \(N\). \(F\)Force vectors of various parts of the system. \(N\)Torque vector of each part of the system. \(q\)A vector composed of \(h\) vectors from multiple terminals. \(O\)Origin of coordinate systems for various parts of the system. \(v\)Translation velocity vector of various parts of the system. \(p\)Position vectors of various parts of the system. \(Q\)Direction vectors of various parts of the system. \(l\)Virtual rod vector from the coordinate origin \(O_{hi}\) of each terminal to the object coordinate origin \(O_{o}\). \(\omega\)Angular velocity vectors of various parts of the system. The following subscripts are defined to distinguish the position, velocity, and force vectors of each part of the system. \(o\)Position, velocity, and force related to objects. \(li\)Position, velocity, and force of the \(i\)th virtual rod tip. \(hi\)Position, velocity, and force at the root of the virtual rod (at the \(i\)th contact terminal). \(hj\)The position and speed of the root of the virtual rod (at the \(i\)th slave terminal) after system simplification. \(i\)Contact point or terminal serial number. \(j\)Represents the simplified serial numbers of each slave terminal, \(j=a,b,c\). \(r\)Relative (internal) position, velocity, and force. \(m\)Generalized relative (internal) position, velocity, and force vectors. \(e\)Measure of position, velocity, and force. The following superscript is defined in the upper left corner of the vector to distinguish the coordinate system represented by the vector. \(w\)Robot coordinate system \(\Sigma_{w}\). \(hi\)Simplified coordinate systems for each terminal \(\Sigma_{hi}\), \(i=a,b,c,d\). \(hj\)Simplified coordinate systems for each slave terminal \(\Sigma_{hi}\), \(j=b,c,d\). Please refer to the text for other symbols not defined here. This paper considers the process of a humanoid remote-operation robot equipped with four end effectors to manipulate objects with unknown information other than visual information. Figure 1 illustrates each terminal, which has two customized optical bionic force sensors. These four terminals cooperate to stably clamp unknown objects and maintain adaptive stable handling without relative sliding during any collaboration of the remote operation or the planning path. This paper makes the following assumptions:  
1. Remote operation enables each force-sensing contact point installed at the terminal to reach a reasonable, cooperative position on the object's surface, and all contact points contact the object's surface. 2. The remote operation causes the four terminals to form an origin \(O_{o}\) in the coordinate system \(\Sigma_{o}\), as illustrated in Figure 1, which is close to the object's center of gravity. A virtual rod defines the force space control vector in a multi-terminal multi-contact-point collaborative system [23]. The contact point of this paper is a three-dimensional optical bionic force sensor (Figure 1). The coordinate system \(\Sigma_{hi}(i=1,\ldots,n)\) represents the contact point coordinate system (\(n=8\)). Nevertheless, the proposed method can be applied to more contact points and terminals. \(\Sigma_{o1}\) and \(\Sigma_{o2}\) represent the planar center coordinate system formed by the four coordinate system origins at the contact points of the two terminals installed on a single robotic arm. In this paper, \(O_{o}\) is set at the midpoint of the connecting line between \(O_{1}\) and \(O_{2}\). The virtual rod \({}^{w}I_{hi}\) from \(O_{hi}\) to \(O_{o}\) is equivalent to a rigid rod structure fixed at the contact point. The virtual rod top coordinate system \(\Sigma_{hi}(i=1,\ldots,8)\) initially coincides with the object coordinate system \(\Sigma_{o}\).