**Abbreviations**

The following abbreviations are used in this manuscript:


#### **Appendix A. Mass-Matrix for Individual Linkages**

It is necessary to define the mass matrix for each of the linkages in order to assemble the mass matrix of the entire mechanism.

Given that each counterweight is firmly attached to its corresponding link and that their position with respect to the local reference system does not change with the mechanism's motion, it is possible to consider five linkages to be optimized, each of them encompassing a link and a counterweight.

Linkages 1 and 3 have mass matrices consisting of three basic points, while linkages 2, 4, and 5 have mass matrices involving two basic points.

To define the different mass matrices, it is necessary to substitute the terms *m*, *<sup>x</sup>*¯*g*, *y*¯*g*, *Ix*, *Iy*, *Ixy*, and *Iz* in the equations that define the mass matrices of two and three points, considering the contribution of both the link and the counterweight. Thus, for each *n* linkage, Equations (A1)–(A7) can be written.

$$m\_n = m\_{bn} + m\_{cn} = m\_{bn} + \rho \pi t\_{cn} (\mathbf{x}\_{cn}^2 + \mathbf{y}\_{cn}^2) \tag{A1}$$

$$\mathbf{x}\_{\mathcal{S}^n} = \frac{\mathbf{x}\_{\text{bul}} m\_{\text{bul}} + \mathbf{x}\_{\text{c}\text{m}} m\_{\text{c}\text{n}}}{m\_{\text{b}\text{m}} + m\_{\text{c}\text{n}}} \tag{A2}$$

$$y\_{\mathcal{S}^n} = \frac{y\_{\text{bıt}} m\_{\text{bıt}} + y\_{\text{cıt}} m\_{\text{cıt}}}{m\_{\text{bıt}} + m\_{\text{cıt}}} \tag{A3}$$

$$I\_{\rm xn} = I\_{\rm xbm} + I\_{\rm xcn} = I\_{\rm xbn} + \frac{1}{4} \rho \tau t\_{\rm cn} (\mathbf{x}\_{\rm cn}^2 + y\_{\rm cn}^2)(\mathbf{x}\_{\rm cn}^2 + 5y\_{\rm cn}^2) \tag{A4}$$

$$I\_{yn} = I\_{ybn} + I\_{ycn} = I\_{ybn} + \frac{1}{4} \rho \tau t\_{cn} (x\_{cn}^2 + y\_{cn}^2)(5x\_{cn}^2 + y\_{cn}^2) \tag{A5}$$

$$I\_{xyn} = I\_{xybn} + I\_{xycn} = I\_{xybn} + \rho \tau t\_{cn} (\mathbf{x}\_{cn}^2 + y\_{cn}^2) \mathbf{x}\_{cn} y\_{cn} \tag{A6}$$

$$I\_{zu} = I\_{zbn} + I\_{zcn} = I\_{xbn} + I\_{ybn} + I\_{xcn} + I\_{ycn} \tag{A7}$$

#### *Appendix A.1. Mass Matrix for Linkages 1 and 3*

To define the mass matrix of linkage 1, let us consider point C as *i*, point E as *j*, and point D as *k*. Similarly, for linkage 3, point B is considered as *i*, point F as *j*, and point E as *k*.

Figure A1 shows the distances *Kx*1, *Ky*1, *Kx*3, and *Ky*3. Note that for linkage 1, *lij* = *l*1 corresponds to the distance *CE* ¯ while for linkage 2, *lij* = *l*2 corresponds to the distance *BF* ¯ .

**Figure A1.** Linkages defined by three basic points. (**a**) Linkage 1. (**b**) Linkage 3.

By using the mass matrix **M3P** (Equation (24)), the terms of the mass matrix for linkages *n* = 1 and *n* = 3 can be defined according to Equations (A8)–(A13).

*en* = *IxbnK*<sup>2</sup>*xn <sup>K</sup>*2*ynl*2*n* − 2*IxbnKxn <sup>K</sup>*2*ynln* + *Ixbn <sup>K</sup>*2*yn* − 2*IxybnKxn Kynl*<sup>2</sup>*n* + 2*Ixybn Kynln* + *Iybn l*2*n* + *<sup>π</sup>K*2*xnρcntcnx*<sup>4</sup>*cn* <sup>4</sup>*K*2*ynl*2*n* + <sup>3</sup>*πK*2*xnρcntcnx*<sup>2</sup>*cn* <sup>2</sup>*K*2*ynl*2*n <sup>y</sup>*<sup>2</sup>*cn* + <sup>5</sup>*πK*2*xnρcntcny*<sup>4</sup>*cn* <sup>4</sup>*K*2*ynl*2*n* + 2*Kxnmbnybn Kynln* + 2*πKxnρcn Kynln tcnx*<sup>2</sup>*cnycn* + 2*πKxnρcn Kynln tcny*<sup>3</sup>*cn* − 2*πKxnρcn Kynl*<sup>2</sup>*n tcnx*<sup>3</sup>*cnycn* − 2*πKxnρcn Kynl*<sup>2</sup>*n tcnxcny*<sup>3</sup>*cn* − *<sup>π</sup>Kxnρcntcnx*<sup>4</sup>*cn* <sup>2</sup>*K*2*ynln* − 3*πKxnρcn <sup>K</sup>*2*ynln tcnx*<sup>2</sup>*cny*<sup>2</sup>*cn* − <sup>5</sup>*πKxnρcntcny*<sup>4</sup>*cn* <sup>2</sup>*K*2*ynln* − <sup>2</sup>*m*2*bnxbn lnmbn* + *<sup>π</sup>lnρcntcnx*<sup>2</sup>*cn* + *<sup>π</sup>lnρcntcny*<sup>2</sup>*cn* − <sup>2</sup>*m*<sup>2</sup>*bnybn Kynmbn* + *<sup>π</sup>Kynρcntcnx*<sup>2</sup>*cn* + *<sup>π</sup>Kynρcntcny*<sup>2</sup>*cn* − <sup>2</sup>*πmbnρcntcnxbnx*<sup>2</sup>*cn lnmbn* + *<sup>π</sup>lnρcntcnx*<sup>2</sup>*cn* + *<sup>π</sup>lnρcntcny*<sup>2</sup>*cn* − <sup>2</sup>*πmbnρcntcnxbny*<sup>2</sup>*cn lnmbn* + *<sup>π</sup>lnρcntcnx*<sup>2</sup>*cn* + *<sup>π</sup>lnρcntcny*<sup>2</sup>*cn* − <sup>2</sup>*πmbnρcntcnx*<sup>3</sup>*cn lnmbn* + *<sup>π</sup>lnρcntcnx*<sup>2</sup>*cn* + *<sup>π</sup>lnρcntcny*<sup>2</sup>*cn* − <sup>2</sup>*πmbnρcntcnx*<sup>2</sup>*cnybn Kynmbn* + *<sup>π</sup>Kynρcntcnx*<sup>2</sup>*cn* + *<sup>π</sup>Kynρcntcny*<sup>2</sup>*cn* − <sup>2</sup>*πmbnρcntcnx*<sup>2</sup>*cnycn Kynmbn* + *<sup>π</sup>Kynρcntcnx*<sup>2</sup>*cn* + *<sup>π</sup>Kynρcntcny*<sup>2</sup>*cn* − <sup>2</sup>*πmbnρcntcnxcny*<sup>2</sup>*cn lnmbn* + *<sup>π</sup>lnρcntcnx*<sup>2</sup>*cn* + *<sup>π</sup>lnρcntcny*<sup>2</sup>*cn* − <sup>2</sup>*πmbnρcntcnybny*<sup>2</sup>*cn Kynmbn* + *<sup>π</sup>Kynρcntcnx*<sup>2</sup>*cn* + *<sup>π</sup>Kynρcntcny*<sup>2</sup>*cn* − <sup>2</sup>*πmbnρcntcny*<sup>3</sup>*cn Kynmbn* + *<sup>π</sup>Kynρcntcnx*<sup>2</sup>*cn* + *<sup>π</sup>Kynρcntcny*<sup>2</sup>*cn* − <sup>2</sup>*π*<sup>2</sup>*ρ*<sup>2</sup>*cnt*2*cnx*5*cn lnmbn* + *<sup>π</sup>lnρcntcnx*<sup>2</sup>*cn* + *<sup>π</sup>lnρcntcny*<sup>2</sup>*cn* − <sup>2</sup>*π*<sup>2</sup>*ρ*<sup>2</sup>*cnt*2*cnx*<sup>4</sup>*cnycn Kynmbn* + *<sup>π</sup>Kynρcntcnx*<sup>2</sup>*cn* + *<sup>π</sup>Kynρcntcny*<sup>2</sup>*cn* − <sup>4</sup>*π*<sup>2</sup>*ρ*<sup>2</sup>*cnt*2*cnx*3*cny*<sup>2</sup>*cn lnmbn* + *<sup>π</sup>lnρcntcnx*<sup>2</sup>*cn* + *<sup>π</sup>lnρcntcny*<sup>2</sup>*cn* − <sup>4</sup>*π*<sup>2</sup>*ρ*<sup>2</sup>*cnt*2*cnx*<sup>2</sup>*cny*3*cn Kynmbn* + *<sup>π</sup>Kynρcntcnx*<sup>2</sup>*cn* + *<sup>π</sup>Kynρcntcny*<sup>2</sup>*cn* − <sup>2</sup>*π*<sup>2</sup>*ρ*<sup>2</sup>*cnt*2*cnxcny*<sup>4</sup>*cn lnmbn* + *<sup>π</sup>lnρcntcnx*<sup>2</sup>*cn* + *<sup>π</sup>lnρcntcny*<sup>2</sup>*cn* − <sup>2</sup>*π*<sup>2</sup>*ρ*<sup>2</sup>*cnt*<sup>2</sup>*cny*5*cn Kynmbn* + *<sup>π</sup>Kynρcntcnx*<sup>2</sup>*cn* + *<sup>π</sup>Kynρcntcny*<sup>2</sup>*cn* + 1 + 5*πρcntcn* <sup>4</sup>*l*2*n <sup>x</sup>*4*cn* + 3*πρcntcn* <sup>2</sup>*l*2*n <sup>x</sup>*<sup>2</sup>*cny*<sup>2</sup>*cn* + *πρcntcny*<sup>4</sup>*cn* <sup>4</sup>*l*2*n* + 2*πρcntcn Kynln x*3*cnycn* + 2*πρcntcn Kynln xcny*3*cn* + *πρcntcnx*<sup>4</sup>*cn* <sup>4</sup>*K*2*yn* + 3*πρcntcn* <sup>2</sup>*K*2*yn <sup>x</sup>*<sup>2</sup>*cny*<sup>2</sup>*cn* + 5*πρcntcn* <sup>4</sup>*K*2*yn <sup>y</sup>*<sup>4</sup>*cn* (A8) *fn* = − *IxbnK*<sup>2</sup>*xn <sup>K</sup>*2*ynl*2*n* + *IxbnKxn <sup>K</sup>*2*ynln* + 2*IxybnKxn Kynl*<sup>2</sup>*n* − *Ixybn Kynln* − *Iybn l*2*n* − *<sup>π</sup>K*2*xnρcntcnx*<sup>4</sup>*cn* <sup>4</sup>*K*2*ynl*2*n* − <sup>3</sup>*πK*2*xnρcntcnx*<sup>2</sup>*cn* <sup>2</sup>*K*2*ynl*2*n <sup>y</sup>*<sup>2</sup>*cn* − <sup>5</sup>*πK*2*xnρcntcny*<sup>4</sup>*cn* <sup>4</sup>*K*2*ynl*2*n* − *Kxnmbnybn Kynln* − *<sup>π</sup>Kxnρcntcn Kynln x*<sup>2</sup>*cnycn* − *<sup>π</sup>Kxnρcntcn Kynln <sup>y</sup>*3*cn* + 2*πKxnρcn Kynl*<sup>2</sup>*n tcnx*<sup>3</sup>*cnycn* + 2*πKxnρcn Kynl*<sup>2</sup>*n tcnxcny*<sup>3</sup>*cn* + *<sup>π</sup>Kxnρcntcnx*<sup>4</sup>*cn* <sup>4</sup>*K*2*ynln* + <sup>3</sup>*πKxnρcntcnx*<sup>2</sup>*cn* <sup>2</sup>*K*2*ynln <sup>y</sup>*<sup>2</sup>*cn* + <sup>5</sup>*πKxnρcntcny*<sup>4</sup>*cn* <sup>4</sup>*K*2*ynln* + *mbnxbn ln* + *πρcn ln tcnx*<sup>3</sup>*cn* + *πρcn ln tcnxcny*<sup>2</sup>*cn* − 5*πρcntcn* <sup>4</sup>*l*2*n <sup>x</sup>*4*cn* − 3*πρcntcn* <sup>2</sup>*l*2*n <sup>x</sup>*<sup>2</sup>*cny*<sup>2</sup>*cn* − *πρcntcny*<sup>4</sup>*cn* <sup>4</sup>*l*2*n* − *πρcntcnycn Kynln <sup>x</sup>*3*cn* − *πρcntcnxcn Kynln <sup>y</sup>*3*cn* (A9)

$$\begin{split} g\_{m} &= \frac{l\_{km}k\eta\_{m}}{\xi\_{y}^{2}l\_{m}}\frac{l\_{m}}{k\_{y}^{2}} - \frac{l\_{m}l\_{m}}{\xi\_{y}^{2}l\_{m}} - \frac{l\eta\_{m}l\_{m}}{\xi\_{y}^{2}l\_{m}} + \frac{\pi K\xi\_{m}\rho\_{m}l\_{m}\pi\lambda\_{m}^{2}}{4\xi\_{y}^{2}l\_{m}} + \frac{3\pi K\xi\_{m}\rho\_{m}l\_{m}\pi\lambda\_{m}^{2}}{2\xi\_{y}^{2}l\_{m}}y\_{cm}^{2} \\ &+ \frac{5\pi K\chi\_{m}\rho\_{m}l\_{m}\chi\_{m}^{4}}{4\xi\_{y}^{2}l\_{m}l\_{m}} + \frac{m\eta\_{m}l\_{m}\eta\_{m}}{K\_{\text{ym}}} + \frac{\pi\rho\_{m}l\_{m}}{K\_{\text{ym}}}t\_{m}\chi\_{m}^{2}{}\_{m} + \frac{\pi\rho\_{m}l\_{m}}{K\_{\text{ym}}}t\_{m}y\_{cm}^{3} - \frac{\pi\rho\_{m}l\_{m}l\_{m}\eta\_{m}}{K\_{\text{ym}}l\_{m}}x\_{cm}^{3} \\ &- \frac{\pi\rho\_{m}l\_{m}\iota\_{m}\iota\_{m}}{K\_{\text{ym}}}y\_{m}^{3} - \frac{\pi\rho\_{m}l\_{m}\iota\_{m}\chi\_{m}^{4}}{4\xi\_{y}^{2}} - \frac{3\pi\rho\_{m}l\_{m}\iota\_{m}}{2K\_{\text{ym}}^{2}}x\_{m}^{2}y\_{cm}^{2} - \frac{5\pi\rho\_{m}l\_{m}l\_{m}}{4K\_{\text{ym}}^{3}}y\_{cm}^{4} \\ &= l\_{m}\frac{l\_{m}\iota\_{m}}{\xi\_{y$$

$$\begin{split} \dot{m}\_{n} &= -\frac{I\_{\rm xm}K\_{\rm xn}}{K\_{\rm ym}^{2}l\_{\rm n}} + \frac{I\_{\rm xyln}}{K\_{\rm ym}l\_{\rm n}} - \frac{\pi K\_{\rm xm}\rho\_{\rm c}t\_{\rm c}x\_{\rm c}^{4}}{4K\_{\rm ym}^{2}l\_{\rm n}} - \frac{3\pi K\_{\rm xm}\rho\_{\rm c}t\_{\rm c}x\_{\rm c}^{2}}{2K\_{\rm ym}^{2}l\_{\rm n}}y\_{\rm c}^{2} - \frac{5\pi K\_{\rm xm}\rho\_{\rm c}t\_{\rm c}y\_{\rm c}^{4}}{4K\_{\rm ym}^{2}l\_{\rm n}} \\ &+ \frac{\pi \rho\_{\rm c}t\_{\rm c}x\_{\rm c}y\_{\rm c}}{K\_{\rm ym}l\_{\rm n}}x\_{\rm c}^{3} + \frac{\pi \rho\_{\rm c}t\_{\rm c}x\_{\rm c}y\_{\rm c}}{K\_{\rm ym}l\_{\rm n}}y\_{\rm c}^{3} \\ &\dots \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \dots \end{split} \tag{A12}$$

$$j\_{\rm in} = \frac{I\_{\rm xbn}}{K\_{\rm yn}^2} + \frac{\pi \rho\_{\rm cn} t\_{\rm cn} x\_{\rm cn}^4}{4K\_{\rm yn}^2} + \frac{3\pi \rho\_{\rm cn} t\_{\rm cn}}{2K\_{\rm yn}^2} x\_{\rm cn}^2 y\_{\rm cn}^2 + \frac{5\pi \rho\_{\rm cn} t\_{\rm cn}}{4K\_{\rm yn}^2} y\_{\rm cn}^4 \tag{A13}$$

#### *Appendix A.2. Mass Matrix for Linkages 2, 4, and 5*

Linkages 2, 4, and 5 are defined by two basic points. For linkage 2, point A is considered as *i* while point C is considered as *j*; for linkage 4, point D is considered as *i* while point G is considered as *j*; for linkage 5, point F is considered as *i* while point G is considered as *j*. Each of them have their origin at point *i* and the x axis directed toward point *j*. For each linkage, *ln* is the distance between points *i* and *j*.

By substituting the corresponding terms in the mass matrix of the elements defined by two basic points *M*2*P* (equation can be found in [38]), it is possible to obtain the terms of the mass matrix of each of these elements consisting of a counterweight and a linkage (Equations (A14)–(A17)).

*an* =*mbn* + *πρcntcnx*2*cn* + *y*2*cn* − 1 *ln*(*mbn* + *πρcntcn*(*x*2*cn* + *<sup>y</sup>*<sup>2</sup>*cn*)) 2.0*mbn* + 2.0*πρcntcnx*2*cn* + *<sup>y</sup>*<sup>2</sup>*cnmbnxbn* + *πρcntcnxcnx*2*cn* + *<sup>y</sup>*<sup>2</sup>*cn* + 1 *l*2*n Ibn* + *πρcntcnx*<sup>2</sup>*cnx*2*cn* + *y*2*cn* + *πρcntcny*<sup>2</sup>*cnx*2*cn* + *y*2*cn* + 0.5*πρcntcnx*2*cn* + *<sup>y</sup>*<sup>2</sup>*cn*<sup>2</sup> (A14) *bn* = 1*ln mbnxbn* + *πρcntcnxcnx*2*cn* + *<sup>y</sup>*<sup>2</sup>*cn* − 1 *l*2*n Ibn* + *πρcntcnx*<sup>2</sup>*cnx*2*cn* + *y*2*cn* + *πρcntcny*<sup>2</sup>*cnx*2*cn* + *y*2*cn* + 0.5*πρcntcnx*2*cn* + *<sup>y</sup>*<sup>2</sup>*cn*<sup>2</sup> (A15) *cn* = 1 *ln*(*mbn* + *πρcntcn*(*x*2*cn* + *<sup>y</sup>*<sup>2</sup>*cn*))−*mbn* − *πρcntcnx*2*cn* + *<sup>y</sup>*<sup>2</sup>*cn mbnybn* + *πρcntcnycnx*2*cn* + *<sup>y</sup>*<sup>2</sup>*cn* (A16)

$$d\_{n} = \frac{1}{I\_{n}^{2}} \left( I\_{bn} + \pi \rho\_{cn} t\_{cn} \mathbf{x}\_{cn}^{2} \left( \mathbf{x}\_{cn}^{2} + \mathbf{y}\_{cn}^{2} \right) + \pi \rho\_{cn} t\_{cn} \mathbf{y}\_{cn}^{2} \left( \mathbf{x}\_{cn}^{2} + \mathbf{y}\_{cn}^{2} \right) + 0.5 \pi \rho\_{cn} t\_{cn} \left( \mathbf{x}\_{cn}^{2} + \mathbf{y}\_{cn}^{2} \right)^{2} \right) \tag{A17}$$
