Figure 2 shows the schematic of the constant-force compliant mechanism that is used in the polishing end effector. Each of the flexible beams is fixed on one end while guided on the other end with unchanged tip angle. A guiding spindle is used in the mechanism to increase its bending stiffness. Since the beams undergo large deflections, CBCM [17] is used to characterize their load-deflection behaviors.

### 2.1 Parametric Design

When designing a constant force mechanism utilizing the buckling characteristics of flexible beams, two performance metrics should be considered. One metric is the variation of the output force *ξ* (magnitude of constant force fluctuation), which is defined as:

$$\xi = \max \left( {\frac{{\left| {F - F_{N} } \right|}}{{F_{N} }}} \right),$$

(1)

where *F* is the actual output force of the mechanism at any position in the operational displacement range and *F*_{N} is the desired output force. The other metric is the operational displacement range *S*, which is defined as a continuous flatten part of the force-displacement curve. Detailed definitions of the two performance metrics can be found in our previous work [16].

The current study aims at designing a constant force mechanism for polishing large-aperture reflective mirrors with freeform surfaces. Generally, the contact force needs to be strictly controlled, which should be no more than 50 N (for thin and brittle lenses, the contact force needs to be further reduced). Thus, we set the target contact force as 40 N. According to the structure of the constant force end-effector, the compliant constant force mechanism provides constant force output with 26 N, and the mechanical structure parts of the movable end of the end-effector provide constant force with 14 N. Moreover, the variation of the force fluctuation should not exceed 10%. For a typical polishing process, an operational displacement range of 10 mm is enough.

To meet the performance metrics given above, we can determine the design parameters of the constant force mechanism using the quick design formulas given in Ref. [16]. The problem is formulated as:

subject to

$$\xi \le 0.1, 0 \le \beta \le \frac{{\uppi }}{2},$$

(3)

where *ξ* is the allowed variation of the output force.

Then, we get the curves of *S*/*L* (*L* is the beam length) versus *β* for *ξ* = 10%, which is shown in Figure 3. To meet the requirement of operational displacement *S*/*L* > 0.16, we should choose the *β* in the range from 35° to 47°. As shown in Figure 3, multiple results meet the design requirements. Due to the compactness requirement of the entire structure, the following parameters are selected

$$\begin{gathered} {\raise0.7ex\hbox{$S$} \!\mathord{\left/ {\vphantom {S L}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$L$}} = 0.2, \hfill \\ \beta = 40^\circ . \hfill \\ \end{gathered}$$

(4)

The nondimensionalized output force can be formulated as

$$\frac{{12N_{b} F_{N} L^{2} }}{{EWT^{3} }} = 40.$$

(5)

By rearranging the above equation, the thickness of the flexible beam can be expressed as

$$T = \left( {\frac{{F_{N} L^{2} }}{{3.03N_{b} EW}}} \right)^{1/3} .$$

(6)

The values of the design parameters are determined as: *N*_{b} = 4, *L* = 60 mm, *W* = 5 mm, *T* = 0.2 mm, and *β* = 40°.

### 2.2 Modeling

To further verify the feasibility of the design given in the last section, a kinetostatic model for the constant-force design is established. Due to the symmetry of the design, only one limb is selected to be modeled as a fixed-guided beam. Figure 4 shows discretization of the fixed-guided beam. The global coordinate frame *xOy* is established with its origin placed at the fixed end of the compliant beam (point *A*) and the positive direction of the *x*-axis along the length of the beam. We divide the fixed-guided compliant beam into *N* elements and model the beam using CBCM.

The parameters are defined as follows: *L* is the length of the compliant beam *AB*, *W* is the width, *T* is the thickness, *E* is the Young’s modulus of the material, and *I* is the area moment of inertia (*I* = *WT*^{3}/12). When deflected, *X*_{o}, *Y*_{o}, and *θ*_{o} are used to represent the tip coordinates and the tip angle, respectively.

According to the boundary conditions, *X*_{o} and *Y*_{o} (the coordinates of the guided end) in Figure 4 can be expressed as:

$$X_{o} = L - d\sin \theta_{\it o} ,$$

(7)

$$Y_{o} = (L - X_{o} ){\tan}\theta_{o} ,$$

(8)

where *d* is the displacement at the guided end of the beam. The tip slope remains constant during the movement of the beam, which is shown in Figure 4. Suppose that the beam is divided into *N* elements with equal length. For the *i*th (1 ≤ *i* ≤ *N*) element, its local coordinate frame (*O*_{i}*x*_{i}*y*_{i}) is attached to and moves along with the free end of (*i* − 1)th element, i.e., node (*i* − 1). Note that the first element was fixed to the ground at node 0 and the free end of the beam is node *N*. We use *P*_{i}, *F*_{i}, and *M*_{i} to represent the axial force, the transverse force and the end moment applied on the *i*th element at node *i* with respect to its local coordinate frame, where \(\Lambda\)_{i}, \(\Delta\)_{i}, and *α*_{i} denote the corresponding axial and transverse deflections and the end slope respectively. The discretization introduces 6*N* intermediate parameters of each element, all of which are normalized with respect to the length of an element as:

$$\begin{gathered} p_{i} = \frac{{P_{i} L^{2} }}{{N^{2} EI}},f_{i} = \frac{{F_{i} L^{2} }}{{N^{2} EI}},m_{i} = \frac{{M_{i} L}}{NEI}, \hfill \\ \lambda_{i} = \frac{{N\Lambda_{i} }}{L},\delta_{i} = \frac{{N\Delta_{i} }}{L},\alpha_{i} = \alpha_{i} . \hfill \\ \end{gathered}$$

(9)

The load-deflection relations of each beam element are expressed as:

$$\left[ {\begin{array}{*{20}c} {f_{i} } \\ {m_{i} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {12} & { - 6} \\ { - 6} & 4 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\delta_{i} } \\ {\alpha_{i} } \\ \end{array} } \right] + \frac{{p_{i} }}{30}\left[ {\begin{array}{*{20}c} {36} & { - 3} \\ { - 3} & 4 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\delta_{i} } \\ {\alpha_{i} } \\ \end{array} } \right] + \frac{{p_{i}^{2} }}{6300}\left[ {\begin{array}{*{20}c} { - 9} & {4.5} \\ {4.5} & { - 11} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\delta_{i} } \\ {\alpha_{i} } \\ \end{array} } \right],$$

(10)

$$\lambda_{i} = \frac{{t^{2} p_{i} }}{12} - \frac{1}{60}\left[ {\begin{array}{*{20}c} {\delta_{i} } & {\alpha_{i} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {36} & { - 3} \\ { - 3} & 4 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\delta_{i} } \\ {\alpha_{i} } \\ \end{array} } \right] - \frac{{p_{i} }}{6300}\left[ {\begin{array}{*{20}c} {\delta_{i} } & {\alpha_{i} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} { - 9} & {4.5} \\ {4.5} & { - 11} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\delta_{i} } \\ {\alpha_{i} } \\ \end{array} } \right],$$

(11)

where *t* is the normalized width of the beam.

Since the fixed-guided compliant beam is divided into *N* beam elements, we have 3*N* element equations. In this case, static balancing equations need to be supplemented. The static balancing equation of the *i*th element with respect to its previous element is formulated as follows:

$$\left[ {\begin{array}{*{20}c} {f_{i - 1}^{^{\prime}} } \\ {p_{i - 1}^{^{\prime}} } \\ {m_{i - 1}^{^{\prime}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ {1 + \lambda_{i} } & { - \delta_{i} } & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {f_{i} } \\ {p_{i} } \\ {m_{i} } \\ \end{array} } \right].$$

(12)

Because the *i*th element goes through a rigid body rotation \({\alpha }_{i-1}\) with the deflection of the (*i*−1)th element, we have

$$\left[ {\begin{array}{*{20}c} {f_{i - 1} } \\ {p_{i - 1} } \\ {m_{i - 1} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\cos \alpha_{i - 1} } & {\sin \alpha_{i - 1} } & 0 \\ { - \sin \alpha_{i - 1} } & {\cos \alpha_{i - 1} } & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {f_{i - 1}^{^{\prime}} } \\ {p_{i - 1}^{^{\prime}} } \\ {m_{i - 1}^{^{\prime}} } \\ \end{array} } \right].$$

(13)

Since the local coordinate frame of the first element coincides with the global coordinate frame of the beam, then we have

$$p_{1} = p_{o} ,f_{1} = f_{o} ,m_{N} = m_{o} .$$

(14)

We use *θ*_{i} to represent the rotation of the coordinate frame of the *i*th element relative to the global coordinate frame, thus:

$$\theta_{1} = 0,\theta_{i} = \sum\limits_{k = 1}^{i - 1} {\alpha_{k} \left( {i = 1,2,3, \cdots ,N} \right)} .$$

(15)

The static balancing equations between the first beam element and the *i*th (2 ≤ *i* ≤ *N*) beam element are written as

$$\left[ {\begin{array}{*{20}c} {\cos \theta_{i} } & {\sin \theta_{i} } & 0 \\ { - \sin \theta_{i} } & {\cos \theta_{i} } & 0 \\ {\left( {1 + \lambda_{i} } \right)} & { - \delta_{i} } & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {f_{i} } \\ {p_{i} } \\ {m_{i} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {f_{1} } \\ {p_{1} } \\ {m_{i - 1} } \\ \end{array} } \right].$$

(16)

In addition, there are 3 constraints related to the geometric compatibility for the whole beam:

$$\left\{ {\begin{array}{*{20}c} {\sum\limits_{i = 1}^{N - 1} {\left[ {\left[ {\begin{array}{*{20}c} {\cos \theta_{i} } & { - \sin \theta_{i} } \\ {\sin \theta_{i} } & {\cos \theta_{i} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {L_{i} \left( {1 + \lambda_{i} } \right)} \\ {L_{i} \delta_{i} } \\ \end{array} } \right]} \right] = \left[ {\begin{array}{*{20}c} {X_{\it {0}} } \\ {Y_{\it {0}} } \\ \end{array} } \right],} } \\ {\theta_{N} + \alpha_{N} = \theta_{\it {0}} .} \\ \end{array} } \right.$$

(17)

In summary, the CBCM model has (6*N* + 3) equations to accommodate the 6*N* intermediate parameters generated in the discretization. For the six parameters *F*_{o}, *P*_{o}, *M*_{o}, *X*_{o}, *Y*_{o}, and *θ*_{o} at the end of the fixed-guided compliant beam, if the last three are known, the first three can be solved, vice versa. The reaction force *F* at the guided end of the compliant beam along its moving direction is:

$$F = P_{o} \cos\theta_{o} - F_{o} \sin\theta_{o} .$$

(18)

By substituting the parameters of the fixed-guided compliant beam into the above formula (the beam material is assumed to be 65Mn Spring Steel whose Young’s modulus *E* = 2.1×10^{11} Pa). To further verify the design results, a finite element model for the constant-force design was established in ABAQUS [18]. The beam was divided into 60 elements, with each element modeled using the B22 element. A gradually increased displacement is applied at the beam tip to observe the change of the output force of the mechanism.

The finite element model successfully captures the rapid increase of the output force at the beginning of deflection followed by an extensive range of constant output force. The finite element results are compared with the results of CBCM in Figure 5. The two curves agree well with each other, with the maximum error less than 5%. The results verify effectiveness of the single compliant beam design.