2.1. Structure of the 9-UPS parallel mechanism

As illustrated in Figure 1, a 9-UPS parallel mechanism is designed as demonstration for this novel type of parallel mechanism. Where U represents the universal joint, S represents the spherical joint, and P represents the prismatic joint, respectively. This parallel mechanism consists of a flexible morphing moving platform, a rigid fixed platform, and 9 rigid UPS limbs. Each limb is made up of one active drive linear actuator P, one passive joint U, and one passive joint S. Different from traditional rigid parallel mechanism, the moving platform of this mechanism is composed of a flexible thin plate, which has a large bending deformation ability and negligible tensile deformation ability. Combined with the 9 redundant actuation, the moving platform can be deformed into complex spatial surface shapes.

Figure 1. Prototype of the 9-UPS parallel mechanism.

2.2. Geometry definition

For the subsequent analysis, the frame and key geometry elements of the parallel mechanism are defined, as shown in Figure 2. Where $\left \lbrace W\right \rbrace$ represents the world fixed frame, which is defined in the origin frame of the external measurement device, for example, a 3D camera. $\left \lbrace F_{i}\right \rbrace$ represents the $i-th$ fixed frame, which is located at the center of joint U and fixed with the rigid platform. The $z$ axis direction is vertically upward, the $y$ axis direction is perpendicular to the axis of linear actuator and the $z$ axis direction, and the $x$ axis direction is obtained by the cross product of the $y$ axis and $z$ axis directions. $n_{i},p_{i},q_{i}$ represent the axis of linear actuator, the center point of $i-th$ joint U, and the center point of $i-th$ joint S, respectively.

Figure 2. Geometry definition of the 9-UPS parallel mechanism.

3.1. Shape description based on 3D TPS algorithm

In this subsection, the research work will focus on shape description of the FMP by using limited control points.

Thin plate spline (TPS) method is an efficient and widely used 2D image registration method. The basic idea of this method is to utilize positions of registration points in two images that need to be registered, and through the correspondence of these registration points, obtain the deformation effect of the entire image [Reference Siarohin, Lathuilière, Tulyakov, Ricci and Sebe38]. The physical meaning of TPS method is to ensure that the deformation object has the minimum bending deformation energy while satisfying that all corresponding registration points can be matched.

In this paper, the 2D TPS method [Reference Bookstein39] is further improved so that it can be used to describe the deformation of 3D objects through limited deformation control points. Here this method is named as 3D TPS. As shown in Figure 3, where $\boldsymbol{q}_{i}^{0}$ represents the control point before deformation, $\boldsymbol{q}_{i}^{1}$ represents the control points after deformation, and $N$ is the number of control points. When the control points move in corresponding ways, the entire thin plate will inevitably be distorted. The purpose of this algorithm is to find a reasonable deformation function $\Phi$ , which can be used to describe the coordinates of any point $\boldsymbol{q}_k$ on the thin plate before deformation and the corresponding coordinates $\Phi [\boldsymbol{q}_k]$ on the thin plate after deformation.

Figure 3. The principle of 3D TPS algorithm.

Above all, two parts are defined to describe the loss function during the deformation process. The first part is the fitting term ${\varepsilon }_{\phi }$ , which is used to describe the overall distance deviation after deforming the control points from source to target. The second part is distortion term ${\varepsilon }_{d}$ , which is used to describe the overall distortion level of the thin plate after deformation. The overall loss function is then defined as

(1) \begin{equation} \varepsilon ={{\varepsilon }_{\phi }}+\lambda{{\varepsilon }_{d}} \end{equation}

where $\lambda$ is the weight coefficient used to describe the rigidity of the thin plate.

For the fitting term ${\varepsilon }_{\phi }$ , it is defined as follows

(2) \begin{equation} {{\varepsilon }_{\phi }}={{\sum \limits _{i=1}^{N}{\left \| \Phi (\mathbf{q}_{\mathrm{i}}^{\mathrm{0}})-\mathbf{q}_{\mathrm{i}}^{\mathrm{1}} \right \|}}^{2}} \end{equation}

The distortion term ${\varepsilon }_{d}$ is defined as follows:

(3) \begin{equation} {{\varepsilon }_{d}}=\iiint _{{{R}^{3}}}{\left [ \left ({{\left(\frac{{{\partial }^{2}}\Phi }{\partial{{x}^{2}}}\right)}^{2}}+{{\left(\frac{{{\partial }^{2}}\Phi }{\partial{{y}^{2}}}\right)}^{2}}+{{\left(\frac{{{\partial }^{2}}\Phi }{\partial{{z}^{2}}}\right)}^{2}} \right )+2\left ({{\left(\frac{{{\partial }^{2}}\Phi }{\partial x\partial y}\right)}^{2}}+{{\left(\frac{{{\partial }^{2}}\Phi }{\partial x\partial z}\right)}^{2}}+{{\left(\frac{{{\partial }^{2}}\Phi }{\partial y\partial z}\right)}^{2}} \right ) \right ]dxdydz}\quad \end{equation}

For the solution of the deformation function $\Phi$ , according to the research of FL Bookstein [Reference Bookstein39], if the total loss function Eq.(1) is minimized, the unique closed-form solution of the deformation function can be derived as

(4) \begin{equation} \Phi ({{\boldsymbol{q}}_{\text{k}}})=\mathbf{M}\cdot{{\boldsymbol{q}}_{\text{k}}}+{{\boldsymbol{m}}_{\text{0}}}+\sum \limits _{i=1}^{N}{\boldsymbol{\omega }_{i}U\left(\left \|{{\boldsymbol{q}}_{\text{k}}}-\boldsymbol{q}_{\text{i}}^{\text{0}} \right \|\right)} \end{equation}

where ${{\boldsymbol{q}}_{k}}={{({{x}_{k}},{{y}_{k}},{{z}_{k}})}^{T}}$ represents arbitrary point on the thin plate before deformation, $\boldsymbol{q}_{i}^{0}$ is the control point before deformation, ${{\mathbf{M}}}={{({\boldsymbol{m}_{1}},{\boldsymbol{m}_{2}},{\boldsymbol{m}_{3}})}}$ is the coefficient matrix, $\boldsymbol{m}_0$ is the constant coefficient vector, and $\boldsymbol{\omega }_i$ is the coefficient vector that measures the influence on the control points.

$U(.)$ is a radial basis function, which indicates how the deformation control points affect the point on the thin plate, and is defined as follows:

(5) \begin{equation} U(r)=\left \{ \begin{aligned} &{{r}^{2}}\log r,\quad r\ne 0 \\[5pt] & 0,\quad \quad \quad r=0 \\[5pt] \end{aligned} \right . \end{equation}

From Eq.(4), we can know that, $\mathbf{M}\in{{R}^{3\times 3}},\,{{\boldsymbol{m}}_{\mathrm{0}}}\in{{R}^{3}},\,{{\boldsymbol{\omega }}_{\mathbf{i}}}\in{{R}^{3}}$ , and the total number of the variables is $(1+3+N)\times 3$ .

The control point before deformation $\boldsymbol{q}_{i}^{0}$ can be written in matrix form, and named as control matrix, which is defined as follows:

(6) \begin{equation} \mathbf{P}=\left [ \begin{aligned} & 1\quad{{x}_{1}}^{0}\quad{{y}_{1}}^{0}\quad{{z}_{1}}^{0} \\[5pt] & 1\quad{{x}_{2}}^{0}\quad{{y}_{2}}^{0}\quad{{z}_{2}}^{0} \\[5pt] & \vdots \quad \,\vdots \quad \quad \vdots \quad \,\,\vdots \\[5pt] & 1\quad{{x}_{N}}^{0}\quad{{y}_{N}}^{0}\quad{{z}_{N}}^{0} \\[5pt] \end{aligned} \right ]\in{{\text{R}}^{\text{N}\times \text{4}}} \end{equation}

The control points after deformation $\boldsymbol{q}_{i}^{1}$ can also be written in matrix form, and named as Landmarks Matrix, which is defined as follows:

(7) \begin{equation} \mathbf{Y}=\left [ \begin{aligned} &{{x}_{1}}^{1}\quad{{y}_{1}}^{1}\quad{{z}_{1}}^{1} \\[5pt] &{{x}_{2}}^{1}\quad{{y}_{2}}^{1}\quad{{z}_{2}}^{1} \\[5pt] & \,\vdots \quad \quad \vdots \quad \quad \vdots \\[5pt] &{{x}_{N}}^{1}\quad{{y}_{N}}^{1}\quad{{z}_{N}}^{1} \\[5pt] & \quad \quad \,\,\mathbf{0} \\[5pt] \end{aligned} \right ]\in{{R}^{(N+4)\times 3}} \end{equation}

Define the matrix $\mathbf{K}$ as the following form

(8) \begin{equation} \mathbf{K}=\left [ \begin{aligned} & U({{r}_{11}})\quad U({{r}_{12}})\quad \cdots \quad U({{r}_{1N}}) \\[5pt] & U({{r}_{21}})\quad U({{r}_{22}})\quad \cdots \quad U({{r}_{2N}}) \\[5pt] & \quad \vdots \quad \quad \quad \vdots \quad \quad \,\,\,\vdots \quad \quad \vdots \\[5pt] & U({{r}_{N1}})\quad U({{r}_{N2}})\quad \cdots \quad U({{r}_{NN}}) \\[5pt] \end{aligned} \right ]\in{{R}^{N\times N}} \end{equation}

where $r_{ij}=\left \| \boldsymbol{q}_{\mathrm{i}}^{\mathrm{0}}-\boldsymbol{q}_{\mathrm{j}}^{\mathrm{0}} \right \|$ represents the distance between two control points.

Define the matrix $\mathbf{L}$ as the following form

(9) \begin{equation} \mathbf{L=}\left [ \begin{aligned} & \mathbf{K}\quad \mathbf{P} \\[5pt] &{{\mathbf{P}}^{\mathbf{T}}}\,\,\,\mathbf{0} \\[5pt] \end{aligned} \right ]\in{{R}^{(N+4)\times (N+4)}} \end{equation}

The parameters to be solved for the deformation function can be written in matrix form $(\boldsymbol{\Omega }\, |{{\mathbf{m}}_{\mathrm{0}}}\,{{\mathbf{m}}_{\mathrm{1}}}\,{{\mathbf{m}}_{\mathrm{2}}}\,{{\mathbf{m}}_{\mathrm{3}}})$ . From Eq.(4) and $N$ groups of corresponding deformation control points, the following Eq. can be derived

(10) \begin{equation} \mathbf{Y}=\mathbf{L}{{(\boldsymbol{\Omega }\,|{{\mathbf{m}}_{\mathrm{0}}}\,{{\mathbf{m}}_{\mathrm{1}}}\,{{\mathbf{m}}_{\mathrm{2}}}\,{{\mathbf{m}}_{\mathrm{3}}})}^{T}}\, \end{equation}

where $\boldsymbol{\Omega }=(\boldsymbol{\omega }_1, \boldsymbol{\omega }_2, \,\cdots, \boldsymbol{\omega }_N,)\in{{R}^{3\times N}}$ .

From Eq.(10), the parameter matrix of the deformation function can be obtained

(11) \begin{equation} {{(\boldsymbol{\Omega }\,|{{\mathbf{m}}_{\mathrm{0}}}\,{{\mathbf{m}}_{\mathrm{1}}}\,{{\mathbf{m}}_{\mathrm{2}}}\,{{\mathbf{m}}_{\mathrm{3}}})}^{T}} = \mathbf{L}^{-1} \mathbf{Y} \end{equation}

The deformation function $\Phi$ can be determined from the parameter matrix in Eq.(11). Furthermore, an arbitrary point coordinate after deformation can be calculated from Eq.(4), and the geometric shape of the deformed thin plate can be described.

3.2. Shape control of the morphing platform

In this subsection, we used an effective shape control approach to deform the morphing platform from source to an arbitrary target surface. Figure 4(a) and Figure 4(b) is the source template surface of the morphing platform and an arbitrary target surface, respectively. For the source surface, a serial of control points $\boldsymbol{q}_i$ is predefined. Shape control of the morphing platform is utilizing the control points to achieve an approximately deformation control to the target surface. And the key work is finding the transformed position of the control points $\boldsymbol{q}_i^{'}$ on the target surface, as shown in Figure 4(b).

Figure 4. Shape control of an arbitrary target surface.

A nonrigid ICP algorithm [Reference Amberg, Romdhani and Vetter40] is used to determine the transformed position of the control points on the target surface. This algorithm uses a locally affine regularization which assigns an affine transformation to each points on the surface and minimizes the difference in the transformation of neighboring points. As shown in Figure 5, the source surface is deformed to the target surface. Accordingly, the transformed control points on the target surface are determined.

Figure 5. Control points calculation based on nonrigid ICP algorithm.

Figure 6(a) and Figure 6(b) is the transformed control points on target surface and the approximate shape description based on the 3D TPS algorithm by using nine transformed control points, respectively. It can be seen from the figure that using a limited number of deformation control points can approximately control a complex space surface. However, to obtain more precise control, increasing the number of control points is necessary.

Figure 6. Approximate shape control of the target surface.

4.1. Bending energy definition

In this subsection, we will focus on the definition of bending energy used to describe the deformation energy of FMP. The deformation ability of the FMP (such as carbon fiber plate) used in this research work is determined by the physical properties of its material. The main deformation form is elastic bending deformation, and the other deformation forms, such as tension and compression deformation, are negligible. The equilibrium state of the FMP after deformation compatibility can be solved using the bending energy method, which corresponds to the minimize bending energy state of the FMP under given constraints.

The bending energy of FMP can be expressed as the square of the curvature multiplied by the stiffness of the material, which is defined as follows:

(12) \begin{equation} {{E}_{bending}}=\iint _{D}{\left(\frac{1}{2}{{\kappa }^{2}}{{h}^{3}}\right)}dA\, \end{equation}

where $\kappa$ represents the curvature, $h$ is the thickness of the plate, respectively.

For an elastic thin plate with uniform thickness, the stiffness coefficient $h^3$ only affects the magnitude of the bending energy and can be regarded as a constant coefficient. Therefore, the bending energy of the thin plate under different deformations is mainly affected by the curvature distribution of the thin plate.

The curvature of the plate described by implicit surface in space is mainly related to its normal vector, gradient, and Hessian matrix. The definition of theses corresponding concepts is as follows:

1) Gradient: represents the change rate and direction at a certain point of the implicit surface equation $f(x,y,z)=0$ , which is defined as follows:

(13) \begin{equation} \nabla f=\left(\frac{\partial f}{\partial x},\,\,\frac{\partial f}{\partial y},\,\frac{\partial f}{\partial z}\right)\, \end{equation}

2) Normal Vector: the normal vector $N$ can be obtained through the negative direction of the gradient, which is defined as follows:

(14) \begin{equation} \mathrm{N}=-\nabla f=-\left(\frac{\partial f}{\partial x},\,\,\frac{\partial f}{\partial y},\,\frac{\partial f}{\partial z}\right)\, \end{equation}

3) Hessian Matrix: is the Jacobian matrix of gradient, describes the second derivative information of the surface function $f(x,y,z)=0$ , and is expressed in the following form:

(15) \begin{equation} \mathbf{H}=\left [ \begin{aligned} & \frac{{{\partial }^{2}}f}{\partial{{x}^{2}}}\quad \frac{{{\partial }^{2}}f}{\partial x\partial y}\quad \frac{{{\partial }^{2}}f}{\partial x\partial z} \\[5pt] & \frac{{{\partial }^{2}}f}{\partial y\partial x}\quad \frac{{{\partial }^{2}}f}{\partial{{y}^{2}}}\quad \frac{{{\partial }^{2}}f}{\partial y\partial z} \\[5pt] & \frac{{{\partial }^{2}}f}{\partial z\partial x}\quad \frac{{{\partial }^{2}}f}{\partial z\partial y}\quad \frac{{{\partial }^{2}}f}{\partial{{z}^{2}}} \\[5pt] \end{aligned} \right ] \end{equation}

4) Curvature: describes the bending and twist level of the surface and can be defined in various forms. In particular, normal curvature is rate at which normal is bending along a given tangent direction

(16) \begin{equation} {{\kappa }_{N}}(X)=\frac{\left \langle df(X),\,dN(X) \right \rangle }{{{\left | df(X) \right |}^{2}}}\, \end{equation}

Figure 7. Normal curvature definition.

Figure 8. Principal curvatures definition.

As illustrated in Figure 7, normal curvature equivalent to intersecting surface with normal-tangent plane and measuring the usual curvature of a plane curve. Among all directions $X$ , there are two principal directions $X_1,X_2$ , where normal curvature has minimum / maximum value (respectively). Corresponding normal curvatures are the principal curvatures, written as $\kappa _1, \kappa _2$ , which can be seen in Figure 8.

Principal curvatures $\kappa _1, \kappa _2$ can be calculated through Hessian matrix Eq.(15)

(17) \begin{equation} \left \{ \begin{aligned} &{{\kappa }_{1}}=\frac{trace(\mathbf{H})+\sqrt{trace{{(\mathbf{H})}^{2}}-4\det (\mathbf{H})}}{2} \\[5pt] &{{\kappa }_{2}}=\frac{trace(\mathbf{H})-\sqrt{trace{{(\mathbf{H})}^{2}}-4\det (\mathbf{H})}}{2} \\[5pt] \end{aligned} \right .\, \end{equation}

where $trace(\mathbf{H})$ represents the trace of Hessian matrix, $det(\mathbf{H})$ represents the determinant of Hessian matrix, respectively.

For the implicit surface equation $f(x,y,z)=0$ , define the deformation bending energy by using $\kappa _1, \kappa _2$ , which is shown as follows:

(18) \begin{equation} {{E}_{bending}}=\iint _{S}{(\kappa _{1}^{2}+\kappa _{2}^{2})}dS\, \end{equation}

where principal curvatures $\kappa _1, \kappa _2$ measures the maximum and minimum bending curvature at a given point on the implicit surface. It is a reasonable way to measure the bending energy of the FMP by integrating the sum of principal curvatures’ squares over the entire surface.

4.2. Deformation estimation using PSO algorithm

In this subsection, we will use the bending energy defined in Eq.(18), to calculate the equilibrium state of the FMP after deformation compatibility. According to the bending energy method, this problem can be understood as finding the deformation form with the smallest bending energy among all the possible deformation form sets that satisfy the boundary constrains. A particle swarm optimization (PSO) algorithm is used to solve this optimization problem.

1) Definition of the optimization problem

The objective function of this optimization problem can be defined as the following form:

(19) \begin{equation} {{E}_{bending}}(\boldsymbol{X})={{\sum \limits _{k=1}^{m}{\left \|{{f}_{\kappa }}(\Phi (\boldsymbol{X},\,{{\boldsymbol{q}}_{k}})) \right \|}}^{2}}dS\, \end{equation}

where $\boldsymbol{X}=\left [ R{{y}_{1}},R{{y}_{2}}\cdots R{{y}_{N}}|R{{z}_{1}},R{{z}_{2}}\cdots R{{z}_{N}}|{{l}_{1}},{{l}_{2}},\cdots{{l}_{N}} \right ]\in{{R}^{3N}}$ represents the input parameters that can determine the control points. $N$ represents the number of control points. $m$ represents the number of points in the morphing platform. $Ry_i, Rz_i, l_i$ represent the rotation angle along axis $y$ , the rotation angle along axis $z$ , and length of linear actuator, respectively. where $\boldsymbol{q}_k=(x_k,y_k,z_k)$ represents arbitrary point on the morphing platform before deformation, and $\Phi (\boldsymbol{X}\boldsymbol{q}_k)$ represents the point after deformation by using deformation function $\Phi$ , respectively, where $f_k(\boldsymbol{q}_k^{\Phi })$ represents the principal curvatures calculation function, according to arbitrary deformed point $\boldsymbol{q}_k^{\Phi }$ on the morphing platform, that calculates its principal curvatures $\kappa =[\kappa _1, \kappa _2]$ through a KNN nearby point search algorithm and Eq.(17).

As shown in Figure 2, control points after deformation $\boldsymbol{q}_i^1$ can be expressed as the following form:

(20) \begin{equation} \boldsymbol{q}_{\mathrm{i}}^{\mathrm{1}}={{\boldsymbol{p}}_{\mathrm{i}}}+\mathbf{R}_{\mathrm{Fi}}^{\mathrm{W}}\cdot \mathbf{Ry}(R{{y}_{i}})\cdot \mathbf{Rz}(R{{z}_{i}})\cdot{{\boldsymbol{n}}_{\mathrm{i}}}^{\mathrm{Fi}}\cdot{{l}_{i}}\, \end{equation}

where $\mathbf{R}_{Fi}^W$ represents the rotation matrix from fixed frame $\{Fi\}$ to world frame $\{W\}$ , $\mathbf{Ry}$ represents rotation matrix along $y$ axis, and $\mathbf{Rz}$ represents rotation matrix along $z$ axis, respectively.

This optimization problem can be defined as the following Eq.(21). By giving the input length $l_i$ of each linear actuator, the optimization problem is to find the optimal input parameters $\boldsymbol{X}$ within the constrained space $R{{y}_{i}}\in [R{{y}_{i}}^{L},R{{y}_{i}}^{U}],\quad R{{z}_{i}}\in [R{{z}_{i}}^{L},R{{z}_{i}}^{U}]$ that obtain the minimum bending energy $E_{bending}$ .

(21) \begin{equation} \begin{aligned} &{{\min }_{\boldsymbol{X}}}\quad f(\boldsymbol{X})={{E}_{bending}}(\boldsymbol{X}),\quad \boldsymbol{X}=[R{{y}_{1}},R{{y}_{2}}\cdots R{{y}_{N}}|R{{z}_{1}},R{{z}_{2}}\cdots R{{z}_{N}}|{{l}_{1}},{{l}_{2}},\cdots{{l}_{N}}]\in{{R}^{3N}}\quad \\[5pt] & s.t.\quad \left \{ \begin{aligned} & R{{y}_{i}}\le R{{y}^{U}} \\[5pt] & R{{y}_{i}}\ge R{{y}^{L}} \\[5pt] & R{{z}_{i}}\le R{{z}^{U}} \\[5pt] & R{{z}_{i}}\ge R{{z}^{L}} \\[5pt] \end{aligned} \right .\quad i=1,2,\cdots N \\[5pt] \end{aligned}\, \end{equation}

2) Optimization problem solving using the PSO algorithm

A PSO algorithm is used to solve the optimization problem defined in Eq.(21). This problem is defined in the $D=3N$ dimensional search space and use M particles for the search. Each particle represents a solution and has the following key parameters.

The position of the $i-th$ particle is expressed as:

(22) \begin{equation} {{\boldsymbol{X}}_{\mathrm{id}}}=({{x}_{i1}},\,{{x}_{i2}},\,\cdots, \,{{x}_{iD}})\, \end{equation}

The velocity of the $i-th$ particle is expressed as:

(23) \begin{equation} {{\boldsymbol{V}}_{\mathrm{id}}}=({{v}_{i1}},\,{{v}_{i2}},\,\cdots, \,{{v}_{iD}})\, \end{equation}

The individual optimal position searched by the $i-th$ particle is expressed as:

(24) \begin{equation} {{\boldsymbol{P}}_{\mathrm{id,pbest}}}=({{p}_{i1}},\,{{p}_{i1}},\,\cdots{{p}_{iD}})\, \end{equation}

The swarm optimal position searched by all particles is expressed as:

(25) \begin{equation} {{\boldsymbol{P}}_{\mathrm{d,pbest}}}=({{p}_{1,gbest}},\,{{p}_{2,gbest}},\,\cdots{{p}_{D,gbest}})\, \end{equation}

The fitness value (optimal objective function value) searched by the $i-th$ particle is $f_p$ , searched by the whole swarm is $f_g$ .

Figure 9. Flowchart of particle swarm optimization algorithm.

The flowchart of solving the optimization problem Eq.(21) by using PSO algorithm is described in Figure 9. For each iteration of the calculation loop, the position and velocity of the particles are updated.

For each iteration process, the distance and direction of next movement are calculated from three parts: the inertia of the particle itself, the optimization position of particle itself, and the optimization position of the entire swarm, which is defined as Eq.(26).

(26) \begin{equation} v_{id}^{k+1}=\omega v_{id}^{k}+{{c}_{1}}{{r}_{1}}(p_{id,pbest}^{k}-x_{id}^{k})+{{c}_{2}}{{r}_{2}}(p_{d,gbest}^{k}-x_{id}^{k})\, \end{equation}

The particle position is updated based on the position of previous iteration and velocity of next iteration.

(27) \begin{equation} x_{id}^{k+1}=x_{id}^{k}+v_{id}^{k+1}\, \end{equation}

Finally, the optimal results ${{P}_{d,gbest}},\,{{f}_{g}}$ that satisfies the optimization stopping criterion is obtained. By using $P_{d,gbest}$ , the control points after deformation can be determined, and deformation of the FMP can be described by the deformation function $\Phi$ introduced in Eq.(4).

5.1. Experimental setup

In order to verify the effectiveness of the proposed approach, a physical prototype of the 9-UPS parallel mechanism is built, as shown in Figure 10. The rigid fixed platform is constructed through a optical platform, and the moving platform is made of a $0.5mm$ thick carbon fiber plate that can undergo a wide range of bending deformation. 9 µ linear actuators (prismatic joint) are used for driving, which are connected to the fixed platform and flexible morphing moving platform through the universal (U) joint and spherical (S) joint, respectively. The numbers of the nine linear actuators are marked in Figure 10. The entire 9-UPS parallel mechanism is fixedly placed within the field of view of a Photoneo 3D camera. The 3D point cloud data of the FMP under different driving inputs is captured and compared with the theoretical calculation result to verify the effectiveness of the proposed approach.

Figure 10. Experimental platform of 9-UPS parallel mechanism.

Figure 11. Frame definition of the experimental platform.

Figure 12. Point cloud data of flexible morphing moving platform captured by 3D camera.

The frame definition of the experimental platform is shown in Figure 11, where $\left \lbrace W\right \rbrace$ represents the world fixed frame and is defined in the origin frame of the 3D camera. $\left \lbrace F_{i}\right \rbrace$ represents the $i-th$ fixed frame, which is located at the center of universal joint and fixed with the rigid platform. The $z$ axis direction is vertically upward, the $y$ axis direction is perpendicular to the axis of linear actuator and the $z$ axis direction, and the $x$ axis direction is obtained by the cross product of the $y$ axis and $z$ axis directions. $R_{y}$ , $R_{z}$ , and $l_{i}$ represent rotation angle along $y$ axis, rotation angle along $z$ axis, and length of $i-th$ linear actuator, respectively, and are defined in the frame $\left \lbrace F_{i}\right \rbrace$ . The nine deformation control points are defined in the world fixed frame and are located in the center of the nine spherical joints.

As shown in Figure 12 is the 3D point cloud data of the flexible morphing moving platform captured by the 3D camera, which will be used as the template before deformation.

5.2. Experiment 1: shape description validation

In this subsection, the effectiveness of the proposed shape description approach based on 3D TPS algorithm is verified. It is verified that the overall deformation of the FMP can be approximately described through limited control points.

In the experiment, first, the point cloud data of the FMP before deformation was captured by a 3D camera, and the coordinates of the control points before deformation were extracted from it. Subsequently, by controlling the length of the 9 linear actuators, the FMP is accordingly controlled to undergo different forms of deformation. Meanwhile, using the 3D camera to capture the point cloud data of FMP, which is Scene 1, Scene 2, and Scene 3. The coordinate information of the control points before and after deformation is shown in Table I. As shown in Figure 11, the nine control points is defined in world fixed frame and is located in the center of the nine spherical joints.

Table I. Control points in different deformation scenes.

As shown in Figure 13, for three arbitrarily selected deformed scenes of morphing platform, the point cloud data captured by the 3D camera, the point cloud data calculated using the 3D TPS algorithm, and the description errors between captured data and calculated data are drawn, respectively, which is shown in Figure 13(a), Figure 13(b), and Figure 13(c), respectively. As can be seen from Figure 13(a) and Figure 13(b), the point cloud data calculated using the 3D TPS algorithm is basically consistent with the point data captured by 3D camera, which means that the overall deformation of the morphing platform can be approximately described by the nine control points. Without a doubt, a more precise shape description can be obtained by increasing the number of the deformation control points.

Figure 13. Shape description validation.

In order to quantitatively describe the deviation between the algorithm calculation and the measured data, the cloud to cloud (C2C) absolute distance between point cloud $\boldsymbol{A}$ and point cloud $\boldsymbol{B}$ is defined as follows.

(28) \begin{equation} d\boldsymbol{A,B}=\left [{{\left \|{{\boldsymbol{a}}_{1}}-\boldsymbol{b}_{1}^{neiber} \right \|}_{2}},\quad{{\left \|{{\boldsymbol{a}}_{2}}-\boldsymbol{b}_{2}^{neiber} \right \|}_{2}},\quad \cdots, \quad{{\left \|{{\boldsymbol{a}}_{M}}-\boldsymbol{b}_{M}^{neiber} \right \|}_{2}} \right ] \end{equation}

where $\boldsymbol{a}_i=(x,y,z)^T$ represents arbitrary point in point cloud $\boldsymbol{A}$ . And $\boldsymbol{b}_i^{neiber}$ represents the corresponding closest point with $\boldsymbol{a}_i$ in point cloud $\boldsymbol{B}$ , which is searched through a KNN nearby point search algorithm.

As shown in Figure 13(c) is the C2C absolute distance between the algorithm calculation result and the measured data. The max and mean value of the C2C absolute distance is shown in Table II, in which the mean error is within $1mm$ and max error is within $4mm$ , indicating that the proposed method has a high description accuracy.

5.3. Experiment 2: deformation estimation validation

In this subsection, we will verify the effectiveness of the proposed deformation estimation approach. By giving the input length value $l_i$ of the nine linear actuators and revolution range of the U joint, the equilibrium state of the FMP is calculated. As shown in Table III is the input values of arbitrary three sets of scenes.

According to Eq.(20), deformed control points $\boldsymbol{q}_i^1$ can be uniquely determined by using any set of input parameters $\boldsymbol{X}=[\boldsymbol{ l, Ry, Rz}]$ in Table III. As shown in Figure 11, $R_{y}$ , $R_{z}$ , and $l_{i}$ represent passive rotation angle along $y$ axis, passive rotation angle along $z$ axis, and actuator length of $i-th$ linear actuator, respectively. Furthermore, the deformation function $\Phi$ can be obtained by using Eq.(4). Subsequently, the deformed point cloud data of the FMP can be calculated by using the deformation function.

According to the bending deformation energy calculation formula defined in Eq.(19), the bending energy of the morphing platform was calculated, the result is shown in Figure 14. The color coordinate represents the principal curvature of the point. The hotter color represents greater degree of bending deformation that occurs in this area. According to the definition in Eq.(19) and Eq.(18), we can get that the bending energy is a dimensionless number. For example, 56.2123 in the figure is obtained by integrating the bending energy of the entire point cloud.

As shown in Figure 15, a PSO algorithm is used to solve the deformation compatibility of the FMP. For each iteration step, calculate the bending energy of the possible deformation form of FMP, which is shown from Figure 15(a) to Figure 15(e). The global optimal solution of the PSO algorithm is used as the equilibrium state of deformation compatibility, which is shown in Figure 15(f), where the FMP has the minimum bending energy at this state.

In the three arbitrary selected scenarios, the equilibrium state of deformation compatibility for the FMP is used as the deformation estimation result. As shown in Figure 16 is a comparison graph of the real data measured by 3D camera and the deformation estimation data. It can be seen from the figure that the estimated deformation and the measured deformation have similar deformation forms, which reveals that the proposed deformation estimation approach can approximately describe the deformation results of this type of parallel mechanism and verifies the effectiveness of proposed approach.

Table II. Shape description error by using C2C absolute distance.

Table III. Input length and revolution range of the 9 linear actuator.

Figure 14. Bending energy calculation of flexible morphing platform.

Figure 15. Deformation compatibility analysis by using PSO algorithm.

Figure 16. Deformation estimation based on optimal bending energy.

Figure 17. Shape control of a parabolic cylindrical surface.

Figure 18. Shape control of an elliptic cone surface.

5.4. Experiment 3: shape control validation

In this subsection, the effectiveness of the shape control approach is verified by using two kinds of target surface: parabolic cylindrical surface and elliptic cone surface, as shown in Figure 17(b) and Figure 18(b).

During the experiments, first, we define the nine control points on the source template surface of the morphing platform, which can be seen in Figure 17(a) and Figure 18(a). Subsequently, according to the nonrigid ICP algorithm used in Subsection 3.2, transformed control points on the target surface were calculated, as shown in Figure 17(b) and Figure 18(b). Hereafter, an approximate shape description based on 3D TPS algorithm by using the transformed control points is achieved. As shown in Figure 17(c) and Figure 18(c), the shape approximated using the 3D TPS algorithm has a high similarity to the target surface. Finally, according to the transformed control points, the input length of each linear actuator is determined, which can be seen in Table IV. The actual deformation forms of the morphing platform under these input lengths were captured by a camera, which are shown in Figure 17(d) and Figure 18(d). It can be seen that the actual deformation form has a high similarity with the target surface, which verifies the effectiveness of the proposed deformation control approach.