## 4 Shaking Force Reduction of the 3 RRR PPM
### Redistribution of the Driving Links' Masses  
A second strategy is to eliminate the variation of the shaking force of the manipulator following a two-stage procedure. The first stage is to redistribute the mass of the driving links in order to get a linear relation between the motion of the end-effector and the motion of the global CoM. This can be done by eliminating all the terms (making them equal to zero) in the matrices--\(C_{B1},C_{B2}\) and \(C_{B3}\)--described in Equations (39), (42) and (45); thus, Equation (48) can be changed to calculate the shaking force as follows:  
\[\mathbf{f}_{sh}=m\ddot{\mathbf{g}}=\mathbf{C}_{\mathrm{C1}}\ddot{\mathbf{q}}_{ \mathrm{C1}}+\mathbf{C}_{\mathrm{C2}}\ddot{\mathbf{q}}_{\mathrm{C2}}+\mathbf{C }_{\mathrm{C3}}\ddot{\mathbf{q}}_{\mathrm{C3}} \tag{53}\]  
Note that, in Equation (53), the terms in matrices \(C_{\mathrm{C1}},C_{\mathrm{C2}}\), and \(C_{\mathrm{C3}}\) are constant. Additionally the motion of points \(C_{1},C_{2}\), and \(C_{3}\) can be expressed in terms of the motion of point \(H\) and the rotation of the mobile platform (body 7). In this way, a linear relation between the motion of the end-effector and the motion of the CoM is achieved. Thus, assigning a \"bang-bang\" profile to the trajectory of the the tool and solving the corresponding inverse kinematics problem, it is possible to determine the motion of the driving motors, a very common problem in robotics. As an advantage of using fully Cartesian coordinates modeling, it can be pointed out that the necessary conditions to redistribute the mass among the driving links are expressed directly in terms of the mass and the location of the CoM of the links 1 through 6. The conditions related to limb 1 are as follows:  
Figure 8: Shaking moment comparison: The blue curve is the shaking moment when the tool is guided with a cycloidal profile. The red curve is the shaking moment when the global CoM is guided with a “bang-bang” profile. \[\left(1-\frac{x_{2}}{l_{2}}\right)m_{2}+\frac{x_{1}}{l_{1}}m_{1} =0 \tag{54}\] \[\left(\frac{y_{2}}{l_{2}}m_{2}-\frac{y_{1}}{l_{1}}m_{1}\right) =0 \tag{55}\]  
The conditions related to limb 2 are as follows:  
\[\left(1-\frac{x_{4}}{l_{4}}\right)m_{4}+\frac{x_{3}}{l_{3}}m_{3} =0 \tag{56}\] \[\left(\frac{y_{4}}{l_{4}}m_{4}-\frac{y_{3}}{l_{3}}m_{3}\right) =0 \tag{57}\]  
Finally, the conditions related to limb 3 are as follows:  
\[\left(1-\frac{x_{6}}{l_{6}}\right)m_{6}+\frac{x_{5}}{l_{5}}m_{5} =0 \tag{58}\] \[\left(\frac{y_{6}}{l_{6}}m_{6}-\frac{y_{5}}{l_{5}}m_{5}\right) =0 \tag{59}\]  
The easiest way to comply with the conditions of Equations (55), (57) and (59) is having the CoM of links 2, 4, and 6 inline between their respective joints. This is the case of the manipulator illustrated in this example; see the parameters in Table 1.  
To comply with the conditions in Equations (54), (56) and (58), it is proposed to leave links 2, 4, and 6 as they are and to relocate the CoM of the driving links 1, 3, and 5 by adding counterweights or by changing their form (redistribution of mass). In this case, all links are equal; thus relocation of the CoM of the driving links can be done alike for the driving links. In this case, the relocation of the CoM in these links is achieved just by moving them in the \(x\) direction, keeping the following relation:  
\[x_{i}m_{i}=-l_{i}\left(1-\frac{x_{j}}{l_{j}}\right)m_{j} \tag{60}\]  
where \(i=1,3,5\) and \(j=2,4,6\). To keep the explanation of the method simple and to not distract attention from its application, the mass of the driving links are maintained in the same value: \(m_{1}=m_{3}=m_{5}=1\) kg. Thus, the new location of the CoM of the driving links can be calculated by substituting values in Equation (60) as follows:  
Figure 8:0  
With these new parameters, a \"bang-bang\" motion profile to the straight line trajectory of the end-effector is imposed from its initial point \(H_{i}\) to its final point \(H_{f}\). Figure 9 shows the trajectories generated for the tool and for the CoM.