Keywords

1 Introduction

Concerning on the human-robot interaction [1], Cartesian compliance can significantly improve the safety and reliability of robot manipulators when performing pick-and-place tasks where large contact force will arise from small position misalignment. Hence, parallel continuum manipulators (PCMs) [2, 3] have been increasingly attracted attentions because of their intrinsic advantages, such as reduction of backlash and simplicity to implement. Comparing to their rigid-body counterparts, this new kind of manipulators has intrinsic structural compliance, which is particularly suitable for human-robot interaction [4]. Furthermore, additional properties, such as variable stiffness and reconfigurable working mode, can also be acquired to enrich the potential applications.

With the diversity of working environment, the demands of reconfiguration to the robot are increasing. How to use one robot to solve multiple conditions becomes a popular direction. Research on reconfigurable mechanisms dates back to the 1990s. Dai proposed a kind of mechanism [5] which can change structure if it was folded based on the research to the decorative gifts. Wohlhart found another kind of mechanism [6] which called kinematotropic mechanism. At present, the reconfigurable mechanism is widely applied in many areas. Carroll from Brigham Young University applied the metamorphic method to the manufacture to simplified manufacturing process [7]. Cui enlarges the working space and flexibility of multifingered hand with reconfigurable palm [8].

Furthermore, variable stiffness is another important characteristic of mechanism because of its advantages in processing and manufacturing, such as shaft-hole assembly [9]. The types of variable stiffness mechanisms (VSMs) can be divided into passive one and active one. Different shapes of spring were applied in the design of passive VSMs [10]. The structure of passive VSMs is simple, but its stiffness is different in the case of different external forces. By contrast, active VSMs provided the adjustment of stiffness which have more application. To realize the active control of stiffness, additional actuators like pneumatic actuators [11] and motors [12] are necessary. Most of existing studies devoted to the combination of variable stiffness unit and rigid body. Thus, the range of compliance is limited by the size of variable stiffness unit.

To combine the stiffness varying and reconfiguration capabilities together, a planar parallel continuum manipulator with the structure of 3-PR\(\mathrm {F}_{lex}\) is proposed in this paper. 2-DOF compound drives are introduced to this manipulator to drive three flexible branch limbs. Three redundant actuators give the ability of variable stiffness at the end-effector. Compared with the current active variable stiffness device, the characteristics of variable stiffness were integrated into the body of manipulator which made the structure more compact. Meanwhile, the manipulator is reconfigurable. Through switching the sequence of the compound PR drives, the workspace for orientation of manipulator can be increased. Based on the approach to large deflection problems using principal axes decomposition [13, 14], the kinematics of the manipulator were analyzed.

2 Mechanism Design

As shown in Fig. 1, the planar parallel continuum manipulator (PPCM) studied in this paper, consists of three identical kinematic limbs, with the structure of ‘PR\(\mathrm {F}_{lex}\)’. Here, ‘P’ and ‘R’ represent the conventional prismatic and revolute joints, respectively. ‘\(\mathrm {F}_{lex}\)’ denotes a slender flexible link undergoing nonlinear large deflection. And the underlined ‘PR’ indicates that both the prismatic and revolute joints in the limbs are actively actuated.

Fig. 1.
figure 1

The structure of the 3-PR\(\mathrm {F}_{lex}\) parallel continuum manipulator.

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In each limb, the slender flexible link is fixedly mounted to the corresponding PR joints (which are termed as compound PR drives hereafter in this paper) and the manipulator’s moving platform at its proximal and distal ends, respectively. Different from its rigid-body counterparts, the mobility of the studied PPCM is realized by means of coupling the large deflections of the flexible limbs, rather than the relative motion of rigid joints. Since the compound PR drives are actuated actively, the position and orientation of the flexible limbs can be controlled accordingly at their proximal ends. As a result, the pose of the manipulator’s end-effector can then be articulated precisely for prescribed tasks, by means of properly actuating the limbs’ compound PR drives simultaneously.

Fig. 2.
figure 2

(a–c) Redundancy in actuation for prescribed end-effector poses. (d–f) Reconfiguration of limb structure for the studied PPCM.

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Note that each compound PR drive has two controlling variables. On the contrary, the moving platform of the manipulator is only capable of generating 3-DOF planar displacements. Consequently, there totally exist six independent input variables to control a 3-DOF output motion. In other words, the studied PPCM possesses redundancy in actuation, which results in multiple solutions to the inverse problems of kinetostatics. As demonstrated in Fig. 2(a–c), given a specific pose for the end-effector, different combinations of inputs can be assigned to the PR drives to articulate the manipulator for the prescribed task. In this paper, the capability of varying the Cartesian stiffness is exhibited through a developed manipulator. In Sect. 4, preliminary experiments are conducted on the prototype to verify this characteristics.

Further, a compact tendon-driven differential-motion mechanism is particularly designed in this paper to actuate the 2-DOF compound PR drives in the manipulator’s flexible limbs. Moreover, the three -PR\(\mathrm {F}_{lex}\) limbs of the studied PPCM are arranged parallel within different planes. Hence, the sequence of the flexible limbs is switchable to each other, such that the structure of limbs in the parallel continuum manipulator can be reconfigured to enlarge the workspace, especially the orientation one, for the end-effector.

As exhibited in Fig. 2(d–f), given three pairs of specific inputs for the compound PR drives, the static equilibrium configuration of the redundant PPCM will be quite different, in the case that the flexible limbs are arranged in variant orders. Therefore, by means of changing the sequence of the limbs, the proposed PPCM can work from one configuration to another. Moreover, owing to the structural compliance of flexible links, the working mode of this redundant PPCM can be reconfigured from one to the other, through a continuous path without causing singularity.

3 Kinetostatics Modeling and Analysis

This section presents the kinetostatics modeling and analysis of the synthesized PPCM, as exhibited in Sect. 2. A discretization-based approach, developed in our prior work [14], is then employed as the framework for elastostatics modeling of flexible links undergoing nonlinear large deflections. Using the proposed method, the kinetostatics analysis of the developed flexible parallel manipulator can be implemented in the categories of kinematics/statics of rigid multi-body systems. Moreover, using the product-of-exponentials (POE) formulation [15], gradient-based searching algorithms can be adopted to identify the corresponding static equilibrium configurations of the studied redundant PPCM efficiently.

3.1 Kinetostatics Modeling of the Flexible Links

Based on the principal-axes decomposition of structural compliance matrix [13], a general approach to the nonlinear large deflection problems of slender flexible links is established in our prior work [14], and has been successfully applied to the kinetostatics modeling, analysis, and control of several PCMs [2, 3, 16].

To some extent, the proposed approach can be regarded as a special kind of finite-element method. As shown in Fig. 3, using this method, a slender flexible link will be discretized into a number of small segments with elasticity. By applying the principal-axes decomposition to the structural compliance matrices of these segments, spatial six-DOF serial mechanisms with rigid bodies and passive elastic joints can then be synthesized to characterize the elasticity. Moreover, for the links with uniform cross-section, the corresponding approximation mechanisms can be reconstructed by two sets of orthogonal joints, which are concurrent at a common point. The geometric distribution of the elastic joints and the corresponding mechanical properties, namely the joint twist \({{\boldsymbol{t}}}_{i}\) and the stiffness constants \(k_{i}\), can be uniquely determined according to the decomposition of the segments’ structural compliance. Please refer to [14] for details.

Fig. 3.
figure 3

Mechanism approximation of slender flexible links.

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Furthermore, these elastic joints could be related to particular effects of the segments’ deflections. First, the three revolute joints correspond to the bending about the section plane, and the torsion along the axis, respectively. Meanwhile, the prismatic joints are associated with the shearing and extension/compression deformations. Particularly, as for slender flexible links, the effects of shearing and elongation can be neglected comparing to the bending and torsion. Thus, in the case of planar mechanisms, only the revolute joint corresponding to bending needs to be taken into account. As a result, a planar n-R hyper-redundant linkage can be obtained to approximate the elastostatics of the slender flexible links. Here, n denotes the number of segments discretized from the link.

Using POE formulation, the forward kinematics of the link’s approximation mechanism can be represented as

$$\begin{aligned} \mathbf{g} _{st}(\boldsymbol{\theta })=\exp (\hat{\boldsymbol{\zeta }}_{1}\theta _{1}) \cdots \exp (\hat{\boldsymbol{\zeta }}_{n}\theta _{n})\,\mathbf{g} _{st,0} \end{aligned}$$
(1)

where \(\hat{\boldsymbol{\zeta }}_{i}\in se(3), \, i=1, \cdots , n\), represent the twists of the elastic joints in the approximation mechanism. \(\boldsymbol{\theta }=\left[ {\theta }_{1}, \cdots , {\theta }_{n}\right] ^{T}\in \mathbb {R}^{n\times 1}\) is the corresponding vector of joint variables. \(\mathbf{g} _{st} \in SE(3)\) is the relative pose of the tip frame \(\left\{ \mathrm {T}\right\} \) with respect to the spatial one \(\left\{ \mathrm {S}\right\} \). And \(\mathbf{g} _{st,0}=\mathbf{g} _{st}({{\boldsymbol{0}}})\) relates to the initial position.

Using such a representation, the kinetostatics model can be established by a combination of the geometric constraint to the tip frame in Cartesian space and the static equilibrium condition of the elastic joints in joint space, as

$$\begin{aligned} {{\boldsymbol{f}}}(\boldsymbol{\theta },{{\boldsymbol{F}}})= \left[ \begin{array}{c} {{\boldsymbol{y}}} \\ \boldsymbol{\tau } \\ \end{array}\right] = \left[ \begin{array}{c} (\mathbf{g} _{st}\mathbf{g} _{\,t}^{-1})^{\vee }\\ \mathbf{K} _{\boldsymbol{\theta }} \boldsymbol{\theta }-\mathbf{J} ^{T}{{\boldsymbol{F}}} \end{array}\right] ={{\boldsymbol{0}}} \end{aligned}$$
(2)

where \(\mathbf{g} _{\,t}\) denotes the target pose for \(\left\{ \mathrm {T}\right\} \). \(\mathbf{K} _{\boldsymbol{\theta }}=\mathrm {diag}(k_{1}, \cdots , k_{n})\in \mathbb {R}^{n\times n}\) is the diagonal matrix of joint stiffness. \({{\boldsymbol{F}}}\in \mathbb {R}^{6\times 1}\) represents the external wrench applied at \(\left\{ \mathrm {T}\right\} \). And \(\mathbf{J} \) is the Jocobian matrix of the approximation mechanism, which can be written in a concise form as

$$\begin{aligned} (\dot{\mathbf{g }}_{st}\mathbf{g} _{st}^{-1})^{\vee }=\mathbf{J} \,\dot{\boldsymbol{\theta }}\;\;\Rightarrow \;\;\mathbf{J} =\left[ \boldsymbol{\xi }_{1},\, \cdots ,\, \boldsymbol{\xi }_{n}\right] \in \mathbb {R}^{6\times n} \end{aligned}$$
(3)

where \(\boldsymbol{\xi }_{i}=\mathrm {Ad}(\exp (\hat{\boldsymbol{\zeta }}_{1}{\theta }_{1})\cdots \exp (\hat{\boldsymbol{\zeta }}_{i-1}{\theta }_{i-1}))\boldsymbol{\zeta }_{i}\) represent the joint twists, in \({6}\times {1}\) vector form with respect to \(\left\{ \mathrm {S}\right\} \), in current configuration.

According to the reciprocity between motion/force transmission, it is known that the transpose of the Jacobian matrix \(\mathbf{J} \) simply corresponds to the inverse force Jacobian. Thus, the second equation in (2) relates to the torque deviations between the motion-induced resistance and those transmitted from the tip frame. Further, a closed-form solution to the gradient of (2) can be derived as

$$\begin{aligned} \boldsymbol{\nabla }=\left[ \begin{array}{cc} \frac{\partial {{\boldsymbol{y}}}}{\partial \boldsymbol{\theta }} \;\;&{}\;\; \frac{\partial {{\boldsymbol{y}}}}{\partial {{\boldsymbol{F}}}} \\ \frac{\partial \boldsymbol{\tau }}{\partial \boldsymbol{\theta }} \;\;&{}\;\; \frac{\partial \boldsymbol{\tau }}{\partial {{\boldsymbol{F}}}} \\ \end{array}\right] =\left[ \begin{array}{cc} \mathbf{J} &{} \mathbf{0} \\ \mathbf{K} _{\boldsymbol{\theta }}-\mathbf{K} _\mathbf{J } \;\;&{}\;\; -\mathbf{J} ^{T} \\ \end{array}\right] \end{aligned}$$
(4)

where \(\mathbf{K} _\mathbf{J }=\frac{\partial }{\partial \boldsymbol{\theta }}(\mathbf{J} ^{T}{{\boldsymbol{F}}})\) is the configuration-dependent item of the system’s overall stiffness. The details can be found in [14].

Fig. 4.
figure 4

Mechanism approximation of the 3-PR\(\mathrm {F}_{lex}\) PPCM.

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3.2 Kinetostatics Models of the 3-PR\(\mathrm {F}_{lex}\) PPCM

As indicated in Sect. 2, the PPCM consists of three limbs with the structure of ‘P RF\(_{lex}\)’. Applying the mechanism approximation to each link, a planar hyper-redundant link with two rigid active joints and n passive elastic ones can be constructed. The kinetostatics models of the flexible limbs can be represented as

$$\begin{aligned} \left\{ \begin{array}{l} \mathbf{g} _{j,st}(\boldsymbol{\theta }_{j})=\exp (\hat{\boldsymbol{\zeta }}_{j,1}\theta _{j,1}) \cdots \exp (\hat{\boldsymbol{\zeta }}_{j,n+2}\theta _{j,n+2})\,\mathbf{g} _{j,0}\\ \boldsymbol{\tau }_{j}=\mathbf{K} _{j,p}\boldsymbol{\theta }_{j,p}-\mathbf{J} _{j,p}^{T}{{\boldsymbol{F}}}_{j}\\ \end{array}\right. ,\;\; j=1,2,3 \end{aligned}$$
(5)

where \(\boldsymbol{\theta }_{j}=[\theta _{j,1}, \cdots , \theta _{j,n+2}]^{T}\in \mathbb {R}^{(n+2)\times 1}\) is the limb’s vector of joint variables, including the compound PR drive. So \(\theta _{j,1}\) and \(\theta _{j,2}\) correspond to the translation and rotation of the active prismatic and revolute joints, respectively, as shown in Fig. 4. Accordingly, \(\hat{\boldsymbol{\zeta }}_{j,1}\) and \(\hat{\boldsymbol{\zeta }}_{j,2}\) relate to the corresponding joint twists. While \(\hat{\boldsymbol{\zeta }}_{j,k}, k\ge 3\), are those discretized from the flexible link. \(\boldsymbol{\theta }_{j,p}=[\theta _{j,3}, \cdots , \theta _{j,n+2}]^{T}\in \mathbb {R}^{n\times 1}\) and \(\mathbf{K} _{j,p}=\mathrm {diag}(k_{j,3}, \cdots , k_{j,n+2})\in \mathbb {R}^{n\times n}\) are the variables and stiffness of the passive joints. Similarly, \(\mathbf{J} _{j,p}=[\boldsymbol{\xi }_{j,3}, \cdots , \boldsymbol{\xi }_{j,n+2}]\in \mathbb {R}^{6\times n}\) is the Jacobian matrix relating the passive joints to the tip frame. Here, \(\mathbf{g} _{j,0}\) denotes the initial pose of \(\left\{ \mathrm {T}\right\} \) with respect to \(\left\{ \mathrm {S}\right\} \) in limb j and \({{\boldsymbol{F}}}_{j}\) is the wrench reacted between the moving platform and the corresponding limb.

In the kinetostatics analysis of the whole manipulator, the target poses \(\mathbf{g} _{j,st}\) and the reaction wrenches \({{\boldsymbol{F}}}_{j}\) need to be identified simultaneously for all limbs. So, extra constraints should be introduced to make the problem deterministic.

In parallel manipulators, the kinematic limbs are geometrically compatible to each others at the moving platform, meanwhile, the end-effector should be in static equilibrium undergoing the external loads and the reaction forces from the limbs. Accordingly, the coupling of the manipulator’s limbs can be represented in light of the geometric constraints and static equilibrium conditions as

$$\begin{aligned} \left\{ \begin{array}{l} \mathbf{g} _{1,st}=\mathbf{g} _{2,st}=\mathbf{g} _{3,st}=\mathbf{g} _{st}\\ {{\boldsymbol{F}}}_{e}-{{\boldsymbol{F}}}_{1}-{{\boldsymbol{F}}}_{2}-{{\boldsymbol{F}}}_{3}={{\boldsymbol{0}}}\\ \end{array}\right. \end{aligned}$$
(6)

where \(\mathbf{g} _{st}\) is the target pose for the tool frame. \({{\boldsymbol{F}}}_{e}\) is the external wrench.

Combining (5) and (6), the system kinetostatics model can be obtained as

$$\begin{aligned} {{\boldsymbol{f}}}(\boldsymbol{\theta },{{\boldsymbol{F}}},\mathbf{g} _{st})= \left[ \begin{array}{c} {{\boldsymbol{X}}}_{j}\\ \boldsymbol{\tau }_{0}\\ \boldsymbol{\tau }_{j}\\ \end{array}\right] = \left[ \begin{array}{c} \left( \ln (\mathbf{g} _{j,st}\,\mathbf{g} _{st}^{-1})\right) ^{\vee }\\ {{\boldsymbol{F}}}_{e}-{{\boldsymbol{F}}}_{1}-{{\boldsymbol{F}}}_{2}-{{\boldsymbol{F}}}_{3}\\ \mathbf{K} _{j,p}\,\boldsymbol{\theta }_{j,p}-\mathbf{J} _{j,p}^{T}\,{{\boldsymbol{F}}}_{j}\\ \end{array}\right] ={{\boldsymbol{0}}}, \quad j=1, 2, 3 \end{aligned}$$
(7)

where \({{\boldsymbol{X}}}_{j}=(\ln (\mathbf{g} _{j,st}\,\mathbf{g} _{st}^{-1}))^{\vee }\in \mathbb {R}^{6\times 1}\) denotes the pose deviation of the tip frame. \(\boldsymbol{\theta }=[\boldsymbol{\theta }_1^T, \,\boldsymbol{\theta }_2^T, \,\boldsymbol{\theta }_3^T]^{T}\in \mathbb {R}^{(3n+6)\times 1}\) and \({{\boldsymbol{F}}}=[{{\boldsymbol{F}}}_1^T, \,{{\boldsymbol{F}}}_2^T, \,{{\boldsymbol{F}}}_3^T]^{T}\in \mathbb {R}^{18\times 1}\) are the system vector of joint variables and reaction wrenches.

In the system kinetostatics model (7), there are totally \(3n+12\) equations in terms of \(3n+18\) unknown variables. It is worth noting that, in the studied planar case, the geometric constraints \({{\boldsymbol{X}}}_{j}\) and the equilibrium condition \(\boldsymbol{\tau }_{0}\) degenerate to 3-dimensional components. As well, the reaction wrenches \({{\boldsymbol{F}}}_{j}\) and the target pose \(\mathbf{g} _{st}\) yield to 3-dimensional variables. Consequently, in order to make the model solvable, some of the unknown variables should be specified in advance.

In the forward kinetostatics problem, the inputs of the compound PR drives are given. Thus, the number of unknown variables decreases to \(3n+12\), which is as same as that of the equations. Then, the corresponding forward model can be directly inherited from (7), by replacing the joint variable \(\boldsymbol{\theta }\) with the passive one \(\boldsymbol{\theta }_{p}=[\boldsymbol{\theta }_{1,p}^T, \,\boldsymbol{\theta }_{2,p}^T, \,\boldsymbol{\theta }_{3,p}^T]^{T}\). Further, the corresponding gradient can be derived in closed-form using the same strategy to (4), given by

$$\begin{aligned} \boldsymbol{\nabla }_{fwd}= \left[ \frac{\partial {{\boldsymbol{f}}}_{fwd}}{\partial \boldsymbol{\theta }_{p}},\, \frac{\partial {{\boldsymbol{f}}}_{fwd}}{\partial {{\boldsymbol{F}}}},\, \frac{\partial {{\boldsymbol{f}}}_{fwd}}{\partial \boldsymbol{\xi }_{st}}\right] = \left[ \begin{array}{ccc} \mathbf{J} _{p} \; &{} \; \mathbf{0} \; &{} \; \mathbf{J} _{\boldsymbol{\xi }_{st}}\\ \mathbf{0} \; &{} \; \mathbf{J} _{{{\boldsymbol{F}}}} \; &{} \; \mathbf{0} \\ \mathbf{K} _{p} \; &{} \; -\mathbf{J} _{p}^{T} \; &{} \; \mathbf{0} \\ \end{array}\right] \end{aligned}$$
(8)

where \(\boldsymbol{\xi }_{st}=(\log (\mathbf{g} _{st}))^{\vee }\in \mathbb {R}^{6\times 1}\) is the twist coordinates of \(\mathbf{g} _{st}\). The components can be derived readily by referring to (4), as \(\mathbf{J} _{p}=\frac{\partial {{\boldsymbol{X}}}}{\partial \boldsymbol{\theta }_{p}}\), \(\mathbf{J} _{\boldsymbol{\xi }_{st}}=\frac{\partial {{\boldsymbol{X}}}}{\partial \boldsymbol{\xi }_{st}}\), \(\mathbf{J} _{{{\boldsymbol{F}}}}=\frac{\partial \boldsymbol{\tau }_{0}}{\partial {{\boldsymbol{F}}}}\), and \(\mathbf{K} _{p}=\frac{\partial \boldsymbol{\tau }}{\partial \boldsymbol{\theta }_{p}}\). The details are not given in the paper due to space limitation.

Then, the forward problem of kinetostatics analysis can be solved efficiently using gradient-based searching algorithms, following the update theme as

$$\begin{aligned} {{\boldsymbol{x}}}_{fwd}^{\,(k+1)}={{\boldsymbol{x}}}_{fwd}^{\,(k)}-(\boldsymbol{\nabla }_{fwd}^{\,(k)})^{-1}\,{{\boldsymbol{f}}}_{fwd}^{\,(k)} \end{aligned}$$
(9)

where \({{\boldsymbol{x}}}_{fwd}=[\boldsymbol{\theta }_{p}^{T},\,{{\boldsymbol{F}}}^{T},\,\boldsymbol{\xi }_{st}^{T}]^{T}\) are the unknown variables to be identified.

Similarly, the inverse kinetostatics model can also be derived directly from (7). In this case, the target pose \(\mathbf{g} _{st}\) is prescribed and the corresponding inputs of the active joints require to be determined. The number of unknown variables becomes \(n+15\), which is still greater the number of equations. As indicated in Sect. 2, this property is caused by the actuation redundancy. Due to space limitation, the details of derivation is not provided in the paper.

4 Experimental Validation

To validate the feasibility of the proposed idea, a prototype of the studied PPCM is developed using easy-to-access materials and simple fabrication methods. Preliminary experiments are conducted and the results demonstrate the manipulator’s capability of variable stiffness and reconfiguration.

Fig. 5.
figure 5

Prototype of the studied PPCM with actuation redundancy.

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4.1 Prototyping and Experimental Setup

As illustrated in Fig. 5, a prototype of the studied PPCM is developed in this section. Slender spring-steel strips are employed as the flexible links. A compact cable-driven differential-motion mechanism, as shown in the figure, is particularly designed to actuate the compound PR drives. To make the limb sequence switchable, the three PR\(\mathrm {F}_{lex}\) limbs are arranged parallel on different planes. The Optitrack motion capture system is employed to measure the end pose.

Two sets of differential-motion cables are arranged parallel. When the two driving wheels rotate in the same direction, it generates a translational motion. Otherwise, it produces a rotation. The geometric parameters of the \(\underline{PR}\) drive are as follows: radius of driving wheel - R, radius of passive unit- r, input angle of two driving wheels - \(\phi _{1}\) and \(\phi _{2}\). Then, the relationship between the input and output prismatic joint \(\theta _{1,j}\) and revolute joint \(\theta _{2,j}\) can be written as

$$\begin{aligned} \left[ \begin{array}{c} \theta _{2,j}\\ \theta _{1,j}\\ \end{array}\right] = \left[ \begin{array}{cc} \frac{R}{2r} &{} \frac{-R}{2r}\\ \frac{R}{2} &{} \frac{R}{2} \end{array}\right] \left[ \begin{array}{c} \phi _{1}\\ \phi _{2}\\ \end{array}\right] \end{aligned}$$
(10)
Fig. 6.
figure 6

Load-induced deflections in horizontal vertical directions.

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4.2 Validation of Variable Stiffness

As introduced in Sect. 2, the Cartesian stiffness can be adjusted in the case that the pose of end-effector is prescribed. To verify the characteristics of variable stiffness, a preliminary force compliant experiment was conducted in the section. Based on three different combination of inputs with the same end pose shown in Fig. 2, their longitudinal and lateral force compliant characteristics were analyzed. Horizontal right and vertical downward external forces were respectively applied to the end-effector corresponding to three different combinations of inputs. Then, the displacement in the direction of the force was recorded.

To keep the load direction precise, the external force is applied via a rope traction. The proximal end of rope is fixed on the mid-point of the end-effector, and the distal end is connected to the load. When the horizontal force is applied, the gravity of load is converted to the horizontal force through the fixed pulley, as shown in Fig. 6. The external force is ranged from 0 to 6N with the increment of 1N. The results are shown in Fig. 6.

Fig. 7.
figure 7

Reconfiguration of different working modes.

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Among three different inputs in Fig. 2(a–c), the first has high stiffness in both lateral and longitudinal directions. The third has relatively high longitudinal stiffness, but low lateral stiffness. While in the second case, the manipulator is of lowest stiffness in both lateral and longitudinal directions.

4.3 Validation of Reconfiguration

As indicated in Sect. 2, the sequence of the compound PR drives is switchable to each other, such that the developed prototype of studied PPCM can reconfigure its working mode to increase the workspace, especially for orientation. To validate this characteristics, different sequence of the PR drives was actuated and the corresponding configurations are measured. As shown in Fig. 7, several key positions are illustrated. First, the moving platform can reach a large rotation to the right when three PR drives are arranged in the order of 3-1-2. Then the pose becomes horizontal with adjusting the order to 1-2-3, but the bending direction of flexible links is not changed. In addition, the pose is kept the same, but the bending direction of middle link is changed. Finally, the pose reached a large rotation to the left by adjusting the order to 2-3-1.

5 Conclusion

In this paper, a novel planar parallel continuum manipulator with actuation redundancy is proposed. Different from the traditional rigid mechanism which driven by the motion of rigid joint, the studied manipulator was articulated by large deflection of the coupled flexible links. Through adding redundant actuators and switching the sequence of three PR drives, the variable stiffness and reconfiguration can be respectively realized. Based on the above concept, the prototype was designed. Combining the experiments and Theoretical analysis, the characteristics of variable stiffness and reconfiguration were validated. The proposed manipulator can meet the demands of diversity working scene.