# Research on Human-Robot Collaboration Method for Parallel Robots Oriented to Segment Docking
## 3 Control Design
### Fractional-Order Variable Damping Admittance Control  
To prevent collisions between cabin segments during docking, real-time adjustments to the robot's motion are imperative. This requires the Stewart platform to exhibit high compliance and rapid responsiveness to interaction forces. While admittance control schemes are prevalent in most HRC studies, their conventional use primarily focuses on avoiding additional energy from affecting the system's controllers. This limitation makes traditional admittance control unable to meet the real-time responsiveness and compliance  
Figure 4: Block diagram of a guide frame-controlled robot. The control system consists of a variable damped admittance control generating a compliance trajectory and a robust motion control system. Figure 3: Schematic of the target point on the end face of the segment. demands of robots during collaboration [23]. Hence, improving traditional admittance control is necessary. The goal of traditional admittance control is to make the robot behave like a specific mass-spring-damping system, enabling it to promptly react to the forces applied by the operator and devise a motion plan based on these external forces. The standard law for classical admittance control is given as follows:  
\[M_{D}(\ddot{\boldsymbol{x}}-\ddot{\boldsymbol{x}}_{c})+\boldsymbol{K}_{D}( \dot{\boldsymbol{x}}-\dot{\boldsymbol{x}}_{c})+\boldsymbol{K}_{P}(\boldsymbol {x}-\boldsymbol{x}_{c})=\boldsymbol{Q}_{i} \tag{1}\]  
where \((\boldsymbol{x},\dot{\boldsymbol{x}},\ddot{\boldsymbol{x}})\) is the state of motion when no external force is applied, \(M_{D}\), \(\boldsymbol{K}_{D}\), \(\boldsymbol{K}_{P}\) are the inertia, damping, and stiffness matrices in the Cartesian coordinate system, and \(\boldsymbol{Q}_{i}\) is interaction force during human-computer collaboration. The Cartesian end-position of the robot is denoted by the unit quaternion \(q=\left\{q_{\eta},\boldsymbol{q}_{z}\right\}\), and \(q_{\eta}\) is the scalar part of \(q\) and \(q_{z}\) is the vector part. The interplay between admittance-controlled motion output and force input relies heavily on admittance parameters. Hence, selecting suitable admittance parameters is crucial in HRC. In many HRC scenarios, stiffness matrices and predefined motion plans are often excluded, offering full control of the robot's actions to the operator. From the operator's perspective, the velocity influenced by the damping value, inversely related to the controller's planned velocity, significantly impacts the collaboration's compliance. This paper proposes variable damping admittance control, which is designed to dynamically adjust adduction control parameters based on the external forces encountered during collaboration. To achieve this, a function governing damping is introduced, as follows:  
\[\alpha=tanh(Q_{i}^{T}\boldsymbol{Q}_{i}) \tag{2}\]  
The function's value is limited within the range [0, 1), where it equals 0 in the absence of force in any direction and incrementally approaches 1 with the gradual increase in interaction force. In assembly collaborative tasks, an operator's intention is commonly classified into swift movements and precise adjustments. The operator will exert a larger interaction force when he seeks a swift motion. The robot should reduce damping to augment motion speed, facilitating quicker task accomplishment. On the other hand, when the operator finetunes the position of the movable cabin segment, the robot should elevate damping to diminish motion speed, facilitating the finetuning operation being conducted by the operator. To accomplish this, a variable matrix is suggested as the damping matrix:  
\[\boldsymbol{K}_{D}(\alpha)=(1-\alpha)\boldsymbol{K}_{D_{c}} \tag{3}\]  
where \(\boldsymbol{K}_{D_{c}}\) is the damping matrix with fixed values; \(\boldsymbol{K}_{D}(\alpha)\) is the damping matrix at the time of collaboration \(\boldsymbol{K}_{D_{c}}\) adjusted to the value of variant \(\alpha\).