2.1 Deployable mechanism modeling

The deployable mechanism discussed here is used to deploy and support antenna panels of a Synthetic Aperture Radar (SAR) and its deployed structure is charactered in Fig.1.

Figure 1.

Schematic Diagram of the deployed supporting mechanism in SAR.

As shown in Fig.1, the deployable mechanism is an articulated spatial linkage that is symmetric in the $O-xy$ plane. Moreover, its unfolding motion is also constrained in the same plane. We care about its angular accuracy about the $o-z$ axis most. Therefore, it is convenient to study its kinematics performance in a plane and its planar representation is give in Fig. 2.

Figure 2.

Planar representation of the deployable mechanism.

In Fig. 2, capital letters denote revolute joints. In order to have determined motion, the mechanism is actuated by a motor at O and three auxiliary torsional springs at B, D, and G, respectively, and they are locked after deployment to support the deployed panels. Two panels are designed to be coplanar when they are completely unfolded. However, in practice, the outer panel can still move freely in a small range after the mechanism is locked due to passive joint clearances and the gravity-free working condition in orbit. Error model of the deployable mechanism regarding revolute clearances in its deployed state is depicted in Fig. 3.

Figure 3.

Error model of the deployable mechanism.

Clearances of the locked joints are not included in the error model. Therefore, we simplified links AB and BC into a single link AC with length $L_1$ , and analogously, we obtain the equivalently simplified links CH and CE. No penetration is considered and the clearance is modeled with the commonly used virtual link. Length of the virtual link is computed as:

(1) \begin{equation} r_c = x, \quad x\in[0,r_b-r_j],\end{equation}

where $r_b$ and $r_j$ are the radii of bearing and journal, respectively.

In the error model, the inner panel is error-free due to the locked O and the outer panel may locate randomly within a small range. The midpoint $O_e$ of IE and the body coordinate system $O_e-x_ey_e$ are used to indicate the pose of the outer panel. Given that the error-free pose and the real pose of the outer panel in global frame $O-xy$ are $\mathbf{g}_0$ and $\mathbf{g}_e$ , respectively, we have

(2) \begin{equation} \mathbf{g}_e = \mathbf{g}_0\circ\exp\left(\begin{array}{c@{\quad}c@{\quad}c} 0 & \!\!-\theta_e & x_e\\ \theta_e & 0 & y_e\\ 0 & 0 & 0 \end{array}\right),\end{equation}

where $\mathbf{g}_0$ and $\mathbf{g}_e$ are elements of Lie group SE(2), and $[x_e,y_e]^\top$ and $\theta_e$ are the positioning error and orientation error, respectively. And the error space of the deployable mechanism is defined as the set of all $[x_e,y_e,\theta_e]^\top$ .

2.2 Error space estimation

Apart from the revolute joint clearances, link length tolerances are also considered to estimate the error space. In practice, the inner panel and outer panel are fixed to metal frames and revolute joints I and E are also connected to the frames. Moreover, the connection positions of joints at the frames are adjustable; therefore, errors of the panels brought by tolerances and assembling process can be calibrated and greatly reduced. Then, tolerances of the panels are excluded in this paper. Only the clearances of rods are considered. A geometrical approach is discussed here for error propagation and accumulation analysis.

As discussed previously, the clearance can be represented by a virtual link with maximum length of $r = r_b-r_j$ . Therefore, geometrically, the journal can move freely inside a disk region. Then, it is convenient to use the disk of radius r to represent the error space of a revolute joint. In this way, taking joint F and link FC as an example, the error propagation and accumulation are modeled, as shown in Fig. 4.

Figure 4.

Error propagation and accumulation in an open loop.

In Fig. 4, the error space of joint F due to clearance is represented by a disk with radius $r_1$ , the length of link FC regarding tolerance is in the interval $[L_2-t_{d2},L_2+t_{u2}]$ . The error space at C can be obtained by translating disk from F to C and Mincowski adding with vector $t_d$ and $t_u$ . In a plane, the translation of a region $\mathcal{M}$ with a vector $\mathbf{t}$ is defined as:

(3) \begin{equation} \mathcal{M}+ \mathbf{t} \doteq \left( \begin{array}{c}x_m\\y_m\end{array}\right) +\mathbf{t},\end{equation}

where $(x_m,y_m)^\top$ is any point in $\mathcal{M}$ .

Since the mobility of revolute joint F is not constrained in an open loop, the final error space of C is obtained by rotating the error region generated previously, and finally a ring region is got, suggesting that point C can locate at any position in that ring region. The quadric expression of a planar circle is used to define $\mathcal{C}_{\mathbf{x}_0,R} $ , as:

(4) \begin{equation} \mathcal{C}_{\mathbf{x}_0,R} \doteq (\mathbf{x}-\mathbf{x}_0)^\top\mathbf{Q}(\mathbf{x}-\mathbf{x}_0) - 1\end{equation}

where $\mathbf{x}$ is any point in a plane, $\mathbf{x}_0$ is the center of a circle, R is the radius of a circle, and $\mathbf{Q}$ is a $2\times2$ diagonal matrix, as:

(5) \begin{equation} \mathbf{Q} = \left(\begin{array}{c@{\quad}c} \dfrac{1}{R^2} & 0\\[5pt] 0 & \dfrac{1}{R^2} \end{array}\right).\end{equation}

Then the error space of C considering clearance at F and tolerance can be expressed as:

(6) \begin{equation} \mathcal{M}_C =\{\mathcal{C}_{\mathbf{x}_C,L_2-r_1-t_{d2}}\geq 0\} \cap \{\mathcal{C}_{\mathbf{x}_C,L_2+r_1+t_{u2}}\leq 0\} \end{equation}

where $\mathbf{x}_c$ denotes the error-free coordinates of C.

Now that the geometric expression of clearance and tolerance propagating and accumulating in the open loop is obtained, and the error space in the closed loop is to be derived next. As shown in Fig. 3, the deployable mechanism owns a multi-closed-loop structure, and the error propagation path should be studied before estimating the final error space of panel IE. Obviously, for the deployable mechanism, error spaces at joints A, F, H, and I are only decided by their own clearances, the error space of C is decided in multiple loops ACF and FCH, and the error space of E is decided by joint C, link CE and its own clearance. Then, the error space of C is geometrically modeled, as shown in Fig. 5.

Figure 5.

Error space of joint C.

In Fig. 5, $\{r_{A1},r_{A2}\},\{r_{F1},r_{F2}\}$ , and $\{r_{H1},r_{H2}\}$ are radii of rings representing the error spaces of A, F, and H, respectively. As discussed previously, error spaces at C in open loops AC, FC, and HC can be represented by rings denoted as $^A\mathcal{M}_C$ , $^F\mathcal{M}_C$ , and $^H\mathcal{M}_C$ , respectively, as shown in Fig. 5. Given that all joints are of the same size $r_1$ and complex revolute joints at C share the common journal that is fixed with link CE, referring to Eq. (6), the propagated error spaces are expressed as:

(7) \begin{equation} \begin{array}{l} ^A\mathcal{M}_C =\{\mathcal{C}_{\mathbf{x}_A,r_{A1}}\geq 0\} \cap \{\mathcal{C}_{\mathbf{x}_A,r_{A2}}\leq 0\}\\[5pt] ^F\mathcal{M}_C =\{\mathcal{C}_{\mathbf{x}_F,r_{F1}}\geq 0\} \cap \{\mathcal{C}_{\mathbf{x}_F,r_{F2}}\leq 0\}\\[5pt] ^H\mathcal{M}_C =\{\mathcal{C}_{\mathbf{x}_H,r_{H1}}\geq 0\} \cap \{\mathcal{C}_{\mathbf{x}_H,r_{H2}}\leq 0\} \end{array}, \end{equation}

where

(8) \begin{equation} \begin{array}{l} r_{A1} = L_1-2r_1-t_{d1}\\[5pt] r_{A2} = L_1+2r_1+t_{u1}\\[5pt] r_{F1} = L_2-2r_1-t_{d2}\\[5pt] r_{F2} = L_2+2r_1+t_{u2}\\[5pt] r_{H1} = L_3-2r_1-t_{d3}\\[5pt] r_{H2} = L_3+2r_1+t_{u3} \end{array}\end{equation}

where $L_1$ , $L_2$ , and $L_3$ are lengths of links AC, FC, and HC, respectively, and their tolerances are $[\!-\!t_{d1},t_{u1}]$ , $[\!-\!t_{d2},t_{u2}]$ , and $[\!-\!t_{d3},t_{u3}]$ , respectively.

Therefore, when considering closed-loop constraints, the error space indicating all the positions journal C can locate is expressed as:

(9) \begin{equation} \mathcal{M}_C = ^A\mathcal{M}_C \cap ^F\mathcal{M}_C \cap ^H\mathcal{M}_C ,\end{equation}

suggesting the intersection region of three rings. Generally, three rings can intersect at up to 24 individual points, and the shape of the error space depends on the intersection situation. Intersection points used for error space computation must satisfy the Eq. (7). As shown in Fig. 5, the error region of C has six intersection points.

As discussed previously, error space of E is computed in the open loop CE, and it consists of the error space propagated from C, error space arisen from length tolerance of $L_4$ and its sole clearance. The propagated part is modeled first, as shown in Fig. 6.

Figure 6.

Error space of joint E.

Despite of closed-loop constraint, the trajectory of E is a circle with radius $L_3$ and center $\mathbf{C}$ . When the error space of C is considered, the error space of E arisen from C can be obtained by translating (without rotation) $\mathcal{M}_C$ to every position on the trajectory of E. The region enclosing all translated $\mathcal{M}_C$ is the error space of E propagated from C, denoted as $^C\mathcal{M}_E$ . However, the contour of $^C\mathcal{M}_E$ is difficult to express with equations. Then, a simplified ring is used to substitute the $^C\mathcal{M}_E$ . Using the $\mathcal{M}_C$ translated to the error-free position of E, the radii of the substituting ring are computed. Given that P is an arbitrary point on $\mathcal{M}_C$ at E, $\mathbf{r}_P$ is a vector from E to P and $\mathbf{r}_{CE}$ is a vector from C to E, then the projection of $\mathbf{r}_P$ on CE is computed as:

(10) \begin{equation} \mathbf{r}_{P1} = \frac{(\mathbf{r}_{P}\cdot\mathbf{r}_{CE})\mathbf{r}_{CE}}{|\mathbf{r}_{CE}|}\end{equation}

Then, the radii of the ring are computed as:

(11) \begin{equation} \begin{array}{l} {r}_{E1} = L_3 + \min({\mathbf{r}_{P1}})\\ {r}_{E2} = L_3 + \max({\mathbf{r}_{P1}}) \end{array} \end{equation}

When the length tolerance $[\!-\!t_{d4}, t_{u4}]$ and clearance of E are included, the finale error space of E is expressed as:

(12) \begin{equation} \mathcal{M}_E = \{\mathcal{C}_{\mathbf{x}_C,{r}_{E1}-r_1-t_{d4}}\geq 0\} \cap \{\mathcal{C}_{\mathbf{x}_C,{r}_{E1}+r_1+t_{u4}}\leq 0\} \end{equation}

Since joint I is fixed to the error-free inner panel, the error space of I is a disk with radius $r_1$ , denoted as $\mathcal{M}_I$ . Then, error space estimation of the outer panel is to find all the possible mobilities of $O_e-x_ey_e$ when two ends of segment IE are strictly restrained in $\mathcal{M}_I$ and $\mathcal{M}_E$ , respectively, as shown in Fig. 7.

Figure 7.

Positioning error space of the outer panel given $\theta = 0$ .

In Fig. 7, $\theta$ is the orientating error. The positioning error space of the outer panel is obtained by translating $\mathcal{M}_I$ and $\mathcal{M}_E$ to the common point $O_e$ and computing the intersected region. A planar disk translated by a vector is expressed as:

(13) \begin{equation} \mathcal{C}_{\mathbf{x}_0,R} + \mathbf{t} \doteq \mathcal{C}_{\mathbf{x}_0+\mathbf{t},R}\end{equation}

Then, the position error space of the outer panel is expressed as:

(14) \begin{equation} \mathcal{M}_P = \{\mathcal{C}_{\mathbf{x}_C+\mathbf{r}_{EO} ,{r}_{d}}\geq 0\} \cap \{\mathcal{C}_{\mathbf{x}_C+\mathbf{r}_{EO},r_u}\leq 0\} \cap \{\mathcal{C}_{\mathbf{x}_I + \mathbf{r}_{IO},r_1} \leq 0\}\end{equation}

Referring to Eq. (12), $r_d = {r}_{E1}-r_1-t_{d4}$ and $r_u = {r}_{E1}+r_1+t_{u4}$ . When angular error is considered, the positioning error space of the outer panel is computed, as shown in Fig. 8.

Figure 8.

Positioning error space of the outer panel given $\theta_e$ .

The positioning error space with angular error $\theta_e$ can be obtained by first rotating vectors $\mathbf{r}_{IO_e}$ and $\mathbf{r}_{EO_e}$ by $\theta_e$ , then translating $\mathcal{M}_I$ and $\mathcal{M}_E$ by rotated vectors $\mathbf{r}_{IO2}$ and $\mathbf{r}_{EO2}$ , respectively, and computing the intersection region. The complete error space of the outer panel is the unite of all the positioning error spaces, modeled as planar regions, of all possible angular errors. It is expressed as:

(15) \begin{equation} \mathcal{M} = \bigcup_{\mathbf{R}_{\theta} \in SO(2)} \{\mathcal{C}_{\mathbf{x}_C+\mathbf{R}_{\theta}\mathbf{r}_{EO} ,{r}_{d}}\geq 0\} \cap \{\mathcal{C}_{\mathbf{x}_C+\mathbf{R}_{\theta}\mathbf{r}_{EO},r_u}\leq 0\} \cap \{\mathcal{C}_{\mathbf{x}_I + \mathbf{R}_{\theta}\mathbf{r}_{IO},r_1} \leq 0\}, \end{equation}

where $\mathbf{R}_{\theta}$ denotes the planar rotation and is expressed as:

(16) \begin{equation} \mathbf{R}_{\theta} = \left(\begin{array}{c@{\quad}c} \cos\theta & -\!\sin\theta\\[5pt] \sin\theta & \cos\theta \end{array}\right).\end{equation}

It is noteworthy that the angular error $\theta$ is usually rather small. When $|\theta|$ is large, we have $\mathcal{M} = \emptyset$ . It is difficult to evaluate Eq. (15) and it can be approximated numerically as:

(17) \begin{equation} \left\{\begin{array}{l} \mathcal{M} = \sum\limits_{i = 1 }^N [ \{\mathcal{C}_{\mathbf{x}_C+\mathbf{R}_{\theta_i}\mathbf{r}_{EO} ,{r}_{d}}\geq 0\} \cap\{\mathcal{C}_{\mathbf{x}_C+\mathbf{R}_{\theta_i}\mathbf{r}_{EO},r_u}\leq 0\} \cap\{\mathcal{C}_{\mathbf{x}_I + \mathbf{R}_{\theta_i}\mathbf{r}_{IO},r_1} \leq 0\} ] \\[5pt] \theta_i = -\pi + \frac{2\pi i}{N} \end{array}\right.,\end{equation}

where N is the number of slices that represent positioning errors under certain orientations.

Given that tolerances are $1/1000$ of link lengths. Parameters used for error space estimation of the deployable mechanism are presented in Table I.

Table I.

Parameter settings.

Figure 9.

Visualization of the final error space.

We sample on x with increment of $0.02$ mm, y with increment of $0.2$ mm, and $\theta$ with increment of $0.02^\circ$ , and the error space of the outer panel is visualized in the three-dimensional Cartesian frame $\{x,y,\theta\}$ , as shown in Fig. 9(a). It is noteworthy that the x-axis, y-axis, and $\theta$ -axis are not in the same scale, so the shape displayed in Fig. 9(a) is deformed.

The error-free pose of the outer panel is given as (850,0,0). It can be observed in Fig. 9 that the positioning error in the x-direction is restrained in [ $-0.52$ mm, $0.5$ mm], the positioning error projected in the x-direction is restrained in [ $-3.33$ mm, $3.31$ mm], and angular error is restrained in $[\!-\!0.764^\circ,0.757^\circ]$ . Moreover, the maximum positioning error is $3.36$ mm at $(849.53,-3.33,-0.714)$ . Since the error space of I (represented by a disk with radius $0.5$ mm) is much smaller than that of E (represented by a ring with width $4.94$ mm), the positioning error is coupled with the angular error, and the translational mobility is mainly determined by $\mathcal{M}_I$ when angular error is fixed, as shown in Fig. 9(g) and (h). The error space of the outer panel is like a cylinder subjected to shear deformation in shape, two ends of which converge to two individual points. In Fig. 9, the translational mobility at any given rotation can be obtained, which gives a deep insight of the accuracy performance of the outer panel.