# Design and Control Simulation Analysis of Tender Tea Bud Picking Manipulator
## 2 Overall Structure Design of Tender Tea Bud-Picking Manipulator
### Manipulator Picking Method and Structural Design
#### 2.2.2 Dynamical Model  
Considering the above proposed 'rotary pull-up' clamping and ripping picking method, a four-degree-of-freedom manipulator can meet the picking needs, and a four-degree-of-freedom spatial manipulator model is established, as shown in Figure 4. Assuming that the mass of the picking manipulator is uniform and friction and nonlinear effects are neglected, the length of the linkage of the robotic arm and the mass of the linkage are \(L_{i}\) and \(m_{i}(i=1,2,3,4)\) in Figure 4. We introduce the Lagrange equation \(\text{K}=\text{T}-\text{V}\) to determine the dynamics model of the system, where T is the kinetic energy of the manipulator and V is the potential energy of the manipulator; the kinematic equation of the manipulator can be expressed as:  
\[\frac{d}{dt}\frac{\partial K}{\partial q}-\frac{\partial K}{\partial q}=\Gamma \tag{1}\]  
where \(\Gamma\) represents the generalized force, since this manipulator has four linkage structures, there are four equations for the generalized coordinate \(q_{i}\)and four equations for the generalized force \(\Gamma_{i}\). Summing the kinetic energy of the four linkage structure links, the total kinetic energy of this manipulator is:  
\[T(\theta,\theta)\sum_{i=1}^{n}T_{i}(\theta,\theta)=\frac{1}{2}\theta^{T}D( \theta)\dot{\theta}\ \ \ \ \ n=4 \tag{2}\]  
where \(D(\theta)\in R^{4\times 4}\) is the inertia matrix of the manipulator. From the Lagrange equation \(K=T-V\), we next require the potential energy of the manipulator arm, so we introduce \(h_{i}\) as the height of the centre of mass of the \(i^{th}\) link.  
\[V_{i}(\theta)=m_{i}gh_{i}(\theta) \tag{3}\]  
Figure 3: Workflow diagram of the premium tea bud-picking manipulator.  
where \(g\) is the Earth's gravitational acceleration and the gravitational component is:  
\[g(\theta)=\frac{\partial V}{\partial\theta_{i}} \tag{4}\]  
where the potential energy \(V\) is:  
\[V(\theta)=g(m_{1}h_{1}(\theta)+m_{2}h_{2}(\theta)+m_{3}h_{3}(\theta)+m_{4}h_{4} (\theta)) \tag{5}\]  
where \(h_{i}\) is the height of the centre of mass of the \(i^{th}\) linkage.  
\[\begin{split} h_{1}(\theta)&=r_{1}\\ h_{2}(\theta)&=L_{1}+r_{2}\sin\theta_{2}\\ h_{3}(\theta)&=L_{1}+L_{2}\sin\theta_{2}+r_{3} \sin(\theta_{2}+\theta_{3})\\ h_{4}(\theta)&=L_{1}+L_{2}\sin\theta_{2}+L_{3} \sin(\theta_{2}+\theta_{3})+r_{4}\sin(\theta_{2}+\theta_{3}+\theta_{4})\end{split} \tag{6}\]  
Finally, bringing Equations (2) and (3) into and combining with Equation (1) yields the Lagrangian dynamics equation of the manipulator as:  
\[K(\theta,\dot{\theta})=\sum_{i=1}^{n}\left(\mathrm{T}_{i}(\theta,\dot{\theta} )-V_{i}(\theta)\right)=\frac{1}{2}\dot{\theta}^{T}D(\theta)\dot{\theta}-V_{i} (\theta) \tag{7}\]  
In order to simplify the equation for ease of solution, the common expression written in matrix form is as follows.  
\[D(\theta)\ddot{\theta}+C(\theta,\dot{\theta})\dot{\theta}+g(\theta)=\tau \tag{8}\]  
where \(D\in R^{4\times 4}\) is the inertia matrix of the manipulator, \(C\in R^{4\times 1}\) is the Coriolis term of the manipulator, \(g(\theta)\in R^{4\times 1}\) is the vector matrix of gravity, and \(\tau\in R^{4\times 1}\) is the moment vector.