## Abstract

Continuum robots have attracted lots of attention due to their structural compliance, manipulation dexterity, and design compactness. To extend the application scenarios, a slender continuum robot, the CurviPicker, was developed for low-load medium-speed pick-and-place tasks in a previous study. To improve the payload capacity and positioning accuracy of the CurviPicker, a novel Continuum Delta Robot (CDR) was then proposed with three dual-continuum-joint translators in a preliminary investigation. However, the initial version of the CDR did not fully utilize the bending ranges of its continuum joints. In addition, while being modeled using the constant curvature assumption for the continuum joints, the CDR shows lowered positioning accuracy for heavier objects, as the CDR’s continuum joints diverge from the assumed constant curvature shapes. In this paper, the design of the CDR was re-optimized to enable wider bending ranges of the continuum joints (>90 deg) to generate an enlarged workspace, taking into consideration several possible structural interferences. Furthermore, a kinetostatic model is derived based on the Cosserat rod theory to reduce the positioning errors caused by the external loads. The experimental result showed that the workspace is enlarged to approximately 9.47 × 10^{7} mm^{3} compared with the volume of 6.57 × 10^{7} mm^{3} of the initial version. Within this enlarged workspace, the average positioning error with a 1000-g load was reduced to 1.93 mm, compared with 4.43 mm obtained by the previous constant curvature assumption.

## 1 Introduction

Due to the inherent compliance, manipulation dexterity, and structural compactness [1,2], continuum robots have been applied to a wide spectrum of medical and industrial tasks, such as in laparoscopic surgery [3,4], natural orifice transluminal endoscopic surgery [5,6], industrial inspection [7–9], and continuum manipulation [10,11].

To expand the application scenarios of continuum robots, the CurviPicker, a slender serial continuum robot, was proposed for pick-and-place tasks [12]. The CurviPicker is composed of an inverted dual-continuum mechanism [3], which generates 2-DoF (degree-of-freedom) translations. However, the CurviPicker is only suitable for low-load medium-speed tasks as the stiffness and payload capability are relatively low, due to its slender shape.

Compared with serial ones, parallel robots have been characterized by generally higher stiffness and accuracy. Hence, parallel continuum robots were proposed for similar purposes. The existing parallel continuum robots are often constructed using continuum joints or flexible rods.

Using continuum joints, both planar and spatial parallel continuum robots were proposed [13–15]. The two planar parallel continuum robots in Refs. [13,15] have three tendon-actuated 1-DoF continuum joints, which are similar to conventional 3-RRR and 3-RRR parallel robots, respectively, where R means the revolute joint. The spatial parallel continuum robot in Ref. [14] consists of three tendon-actuated 2-DoF continuum joints, where each continuum joint connects the moving platform and an active prismatic joint.

Using flexible rods, a spatial parallel continuum robot [16,17] was proposed. The structure is similar to the Stewart–Gough platform. By pushing and pulling the flexible rods, the moving platform can realize 6-DoF motions via the rods’ coupled deformation. Furthermore, a parallel continuum robot was proposed [18], with a passive spring backbone carrying disks that constrain the legs at intermediate points. In addition, a parallel continuum robot [19] was used to actuate the outer sheath by rotating two precurvature super-elastic tubes in single-access minimally invasive procedures. Furthermore, an underactuated 3-PCC (P and C denote the prismatic joint and the continuum joint, respectively) spatial parallel continuum robot was proposed in Ref. [20], by using one motor to actuate three prismatic joints to realize 6-DoF motion, in which two compliant rods in each leg are not coupled. Moreover, these rod-driven parallel continuum robots can also be designed in a reconfigurable topology [21], where a continuum reconfigurable incisionless surgical parallel robot (CRISP) was assembled inside a patient’s abdomen by snaring the flexible tools using snare wires held on snare needles. This parallel continuum robot is actuated by manipulating the snare needles and tools using extracorporeal manipulators.

In our preliminary investigation, a novel parallel continuum robot, named CDR, was proposed with an aluminum base structure, three sliders, three kinematic chains, and a moving platform [22], as shown in Fig. 1(a). Each kinematic chain consists of an active prismatic joint and two passive continuum joints. And it is referred to as a PCC chain. Compared with universal or spherical joints, the continuum joint has a larger angle range. Each continuum joint is a multi-backbone continuum structure, and the backbones in the PJ (proximal joint) are connected to those in the DJ (distal joint) via a rigid multi-lumen tube in between. When an external wrench is exerted on the DJ, the lengths of its backbones will be changed. As the backbones are attached and fixed to the ends of the PJ and the DJ, the bending of the DJ will further induce the bending of the PJ with an identical bending angle in the opposite direction within the same bending plane [3]. Therefore, the end disk of the DJ retains parallel to the end disk of the PJ, referring to Fig. 5. Hence, the structure with the PJ and the DJ is called a dual-continuum-joint translator.

Each chain of the parallel continuum robots in Refs. [13,15] is composed of one planar continuum segment, which cannot realize spatial movement. Compared with the design with six DoFs in Ref. [17], the CDR only has translational movements. And the design in Ref. [20] has possibly undesired non-negligible parasite movements (e.g., tilting of the moving platform). Furthermore, the CDR has a relatively big workspace compared with the other existing parallel continuum robots. Hence, it becomes a promising potential for pick-and-place applications, compared with the related designs of other parallel continuum robots

The initial design in Ref. [22] did not fully utilize the bending capability of the continuum joints, due to an improperly formulated optimization for the structure parameters. Thus, an improved version of the CDR was fabricated in this paper as pictured in Fig. 1(b), where the structural parameters are re-optimized to generate an enlarged workspace.

Furthermore, the constant curvature assumption [23] adopted in the kinematic modeling in the preliminary investigation [22] will lead to large positioning errors, when the continuum joints’ shapes diverge from circular ones, as illustrated in Fig. 1(b). This shape discrepancy is mainly due to the internal wrench exerted on one translator by the other two translators, when the three translators are being deformed in a compatible way to generate movements. And the positioning accuracy can be even lower when external loads are applied. To increase the modeling accuracy for the continuum joints, there are several approaches by considering the material mechanics and the static equilibriums, such as the Cosserat rod theory [16], Euler–Bernoulli beam theory [24], finite element method [25], or the lumped parameter model [26]. The kinetostatic model for the CDR is derived in this paper based on the Cosserat rod theory, which is often used for the kinetostatic modeling of elastic rods and parallel continuum robots [17,27]. In this model, each continuum joint is simplified into a planar bending equivalent rod, in order to the computation and convergence.

Based on the Cosserat rod theory, a kinetostatic model for parallel continuum robots with several intermediate constraints was proposed in Ref. [18]. However, even if the number of intermediate constraints is assumed to be one for each CDR’s continuum joint, the kinetostatic model still has a large computation cost in the preliminary work [28]. In addition, the kinetostatic models proposed in Refs. [16,17,20,21] modeled each flexible rod as a spatial Cosserat rod, also resulting in a lower computation efficiency.

In this work, by utilizing the coupling of two continuum joints of each translator in the presented CDR and simplifying each continuum joint into one planar Cosserat rod, the computation efficiency for the CDR's kinetostatic model can be improved.

The optimization design concepts, kinetostatic model, and experimental characterizations are elaborated. The contributions of this paper are summarized as follows.

The structural parameters of the CDR are re-optimized to generate an enlarged workspace, taking into consideration several possible structural interferences, in order to fully utilize the bending angle ranges of the continuum joints (>90 deg).

A kinetostatic model of the CDR based on the Cosserat rod theory is proposed and validated with the experiments under both the no-load and the loaded conditions.

A series of pick-and-place tasks are conducted to illustrate the effectiveness of the CDR and the kinetostatic model.

## 2 Structure and Re-Optimization of the Continuum Delta Robot

The CDR’s structure is shown in Fig. 1, where the PJs’ end disks of three translators are parallel and fixed on the sliders of three prismatic joints, while their DJs’ end disks are connected to a moving platform. Each continuum joint (the PJ or the DJ) in a translator consists of an end disk, several backbones, and a few spacer disks. The backbones are made from super-elastic nitinol alloy with a diameter of *d*_{Ni-Ti}. The backbones are arranged evenly along a pitch circle whose radius is *r*_{pc}. The backbones can slide freely inside the holes of the spacer disks and the multi-lumen tube, while their ends are fixed at the end disks of the PJ and the DJ.

The continuum joints have large bending angle ranges that are usually larger than those of spherical joints or universal joints (e.g., 90 deg vs. less than 45 deg). While the initial version of the CDR did not fully utilize this feature of the continuum joints, the structural optimization is re-formulated to maximize the workspace of the CDR, while allowing the continuum joints to bend more than 90 deg.

### 2.1 Nomenclature and Coordinate Systems.

Referring to Fig. 2, the nomenclature for the CDR is defined in the appendix and the coordinates for the CDR are listed as follows.

*Reference Coordinate*{*O*} ≡ {$x^$, $y^$_{O}, $z^$_{O}} is attached to the intersection of the guideways’ axes of three prismatic joints. Its_{O}*XY*plane is aligned with the end disk of each PJ and $x^$points to the first translator._{O}*Proximal End Disk Coordinate*{*ipe*} ≡ {$x^$, $y^$_{ipe}, $z^$_{ipe}} is fixed at the center of the PJ’s end disk of the_{ipe}*i*th translator. {*ipe*} is translated along the*i*th guideways’ axes from {*O*} with the actuation value of*q*._{i}*Bending Plane Coordinate*1 of the*i*th translator {*i*1} ≡ {$x^$_{i}_{1}, $y^$_{i}_{1}, $z^$_{i}_{1}} shares its origin with {*ipe*}. Its XY plane coincides with the*i*th translator’s bending plane and $x^$_{i}_{1}is aligned with $z^$._{ipe}*Bending Plane Coordinate*2 of the*i*th translator {*i*2} ≡ {$x^$_{i}_{2}, $y^$_{i}_{2}, $z^$_{i}_{2}} is attached to the center of the DJ’s end disk of the*i*th translator. The*XY*plane of {*i*1} is the same as that of {*i*1} and $x^$_{i}_{2}is perpendicular to the DJ’s end disk of the*i*th translator.*Distal End Disk Coordinate*{*ide*} ≡ {$x^$, $y^$_{ide}, $z^$_{ide}} shares its origin with {_{ide}*i*2}. $x^$points from the center of the CDR’s moving platform to the first translator. $z^$_{ide}is perpendicular to the DJ’s end disk of the_{ide}*i*th translator.*Moving Platform Coordinate*{*P*} ≡ {$x^$, $y^$_{P}, $z^$_{P}} is attached to the center of the CDR’s moving platform. {_{P}*P*} is translated from {*ide*}. Due to the coupled bending of the DJ and the PJ of each translator, {*i*2} is translated from {*i*1}, {*ipe*} is translated from {*ide*}. Thus, {*P*} is translated from {*O*}.

### 2.2 Structural Re-Optimization.

*L*and the inclination of guideway angle

*α*. The total length of each translator is set to

*L*

_{trans}= 2

*L*+

*D*= 500 mm. Then, for an increased

*L*, the length of the multi-lumen tube is shortened, which may make the volume decrease. With the increasing of

*α*, the volume of the CDR’s workspace is indeed expanded. Thus, an optimization problem is formulated in Eq. (1), whose variables are

*α*and

*L*, to make the workspace volume

*V*

_{CDR}as large as possible

*V*

_{CDR}is calculated by scanning an initial volume

**Ω**

_{initial}in {

*O*}, which is big enough to cover all possible positions of the CDR’s moving platform. Discrete positions in this initial volume are checked whether the resolved actuation values are in their ranges (

*q*in [

_{i}*q*

_{i}_{,min},

*q*

_{i}_{,max}]), using the CDR’s inverse kinematics, referring to Ref. [22]. And

*V*

_{CDR}=

*V*

_{Ω}_{initial}·

*n*

_{valid}/

*n*

_{whole}, where

*V*

_{Ω}_{initial}is the volume of

**Ω**

_{initial},

*n*

_{valid}and

*n*

_{whole}are the numbers of the valid positions and the whole positions in

**Ω**

_{initial}, respectively. Using the inverse kinematics, the actuation values

**q**and the bending state of the

*i*th translator (which is represented by (

*θ*,

_{i}*δ*)) can be relatively easily obtained

_{i}*(*

_{CC}

^{O}**p**

*) represents the CDR’s inverse kinematics based on the constant curvature assumption as in Ref. [22].*

_{p}*i*th prismatic joint needs to be set as in Eq. (3)

*d*

_{op,min}is the minimum distance between $z^$

*and $z^$*

_{ipe}*, referring to Fig. 3.*

_{O}*q*=

_{i}*q*

_{i}_{,min}while

*q*=

_{j}*q*

_{j}_{,max}(

*j*= 1, 2, 3 and

*j*≠

*i*), is used to check these possible collisions. And this extreme configuration in the optimization process is identified when |

*q*−

_{i}*q*

_{i}_{,min}| <

*q*

_{thred}and |

*q*−

_{j}*q*

_{j}_{,max}| <

*q*

_{thred}(

*j*= 1, 2, 3 and

*j*≠

*i*), where the threshold

*q*

_{thred}is set as 10 mm in this paper. Referring to Fig. 3, the first possible collision at this extreme configuration is between the PJ of the

*i*th translator and the CDR’s base structure. To avoid the collision, the minimum distance

*d*

_{i}_{tb,min}between the PJ of the

*i*th translator and the base structure needs to be larger than the radius of the

*i*th translator

*r*

_{i}_{PJ}and

^{O}**c**

_{i}_{PJ}are the radius and the center of the PJ’s arc in the

*i*th translator,

^{O}**p**

_{i}_{b}is the edge point of the guideway of the

*i*th prismatic joint, and

*d*

_{trans}is the diameter of each translator.

*r*

_{i}_{PJ},

^{O}**c**

_{i}_{PJ}, and

^{O}**p**

_{i}_{b}are calculated in Eq. (5)

**R**

_{z}(

*θ*) ∈

*SO*(3) is the rotation matrix about z-axis.

*d*

_{tbp}is the outer diameter of each timing belt pulley.

^{O}**p**

_{1b}

^{O}**p**

_{2b}

^{O}**p**

_{3b}. The constraint for this collision is written in Eq. (6)

*d*

_{j}_{mb}is the distance between

^{O}**p**

_{j}_{b}and the centerline of the

*i*th translator’s multi-lumen tube, and

^{O}**p**

_{i}_{mp}and the distal position

^{O}**p**

_{i}_{md}of the

*i*th translator’s multi-lumen tube are calculated as follows:

If *θ _{i}* = 0,

^{O}**p**

_{i}_{mp}=

^{O}**p**

*+ [0 0*

_{ipe}*–L*]

*, and*

^{T}

^{O}**p**

_{i}_{md}=

^{O}**p**

_{i}_{mp}+ [0 0

*–D*]

*.*

^{T}*θ*

_{i,}_{max}of the PJ or the DJ can be calculated in Eq. (9)

*σ*

_{max}is the safety limit of the backbone strain. According to Ref. [29], the safety limit

*σ*

_{max}= 2.5% (an experimental margin of safety to reduce the effects of loading/unloading hysteresis in super-elastic nitinol).

To guarantee that the initial volume **Ω**_{initial} covers the workspace of the CDR, the initial volume **Ω**_{initial} is chosen as a cuboid of 2*L*_{trans} × 2*L*_{trans} × 1.5*L*_{trans}, as shown in Fig. 4(a). The center of **Ω**_{initial} is located on the axis $z^$* _{O}* and the distance between the top plane of

**Ω**

_{initial}and

^{O}**p**

_{1pe}(

*q*

_{1,min}) is 0.5

*L*

_{trans}, where

^{O}**p**

_{1pe}(

*q*

_{1,min}) means the position of {1

*pe*} when

*q*

_{1}=

*q*

_{1,min}.

Equation (10) is solved through a grid-searching strategy with the interval of *α* and *L* designated as 1 deg and 1 mm from 45 deg to 85 deg, and from 40 mm to 80 mm, respectively. As shown in Fig. 4(b), the optimized parameters are *α* = 71 deg and *L* = 60 mm, while *α* = 70 deg and *L* = 60 mm are selected for the convenience of manufacture. To illustrate the optimization clearly, an *L*–*V*_{max} curve is plotted as shown in Fig. 4(c), where *V*_{max} indicates the maximum volume of the workspace of the CDR with a given *L*. Only one solution is to be chosen as with *L* = 60 mm, which is the maximum value on the curve. The detailed structural parameters of this improved version of the CDR are listed in Table 1, where the parameters apart from *α* and *L* are fixed. The optimized actuation range of each translator is that *q _{i}* ∈ [190 mm, 490 mm]. The maximum bending angle of each continuum joint and the workspace volume for this version are 94.59 deg and 9.47 × 10

^{7}mm

^{3}, respectively, compared with the initial ones (86.61 deg and 6.57 × 10

^{7}mm

^{3}). When the length of multi-lumen tube

*D*or the inclination of guideway angle

*α*increases, the workspace of the CDR increases. However, the stiffness of the CDR can decrease with the increase of

*α*.

α = 70 deg | L = 60 mm | ||r|| = 20 mm_{i} | D = 380 mm |
---|---|---|---|

β = _{1}ϕ = 0 deg_{1} | β = _{2}ϕ_{2} = 120 deg | β = _{3}ϕ = 240 deg_{3} | |

d_{tbp} = 38 mm | d = 22 mm_{t} | d_{Ni-Ti} = 1.2 mm | |

r_{pc} = 9 mm | d_{op}_{,}_{min} = 65 mm | q_{range} = 300 mm | |

E = 62 GPa | k_{1} = k_{2} = k_{3} = 5 |

α = 70 deg | L = 60 mm | ||r|| = 20 mm_{i} | D = 380 mm |
---|---|---|---|

β = _{1}ϕ = 0 deg_{1} | β = _{2}ϕ_{2} = 120 deg | β = _{3}ϕ = 240 deg_{3} | |

d_{tbp} = 38 mm | d = 22 mm_{t} | d_{Ni-Ti} = 1.2 mm | |

r_{pc} = 9 mm | d_{op}_{,}_{min} = 65 mm | q_{range} = 300 mm | |

E = 62 GPa | k_{1} = k_{2} = k_{3} = 5 |

## 3 Kinetostatic Modeling for the Continuum Delta Robot

### 3.1 Model Assumptions and Derivations.

In this paper, the kinetostatic model for the CDR is derived based on the Cosserat rod theory. In order to expedite the computation and convergence for this model, a few assumptions are made as follows.

*Assumption*1.*Constant Backbone Lengths*: The lengths of each translator and the backbones inside it are constant, no matter how the continuum joint (the PJ or the DJ) is bent. Thus, the bending angles of each translator’s PJ and DJ are identical but in the opposite direction, when external lateral forces and/or moments are not too big. Please noted that the shape of the PJ and the DJ may be different from each other.*Assumption*2.*Equivalent Rod*: As illustrated in Fig. 5, due to the small spacing between the spacer disks, the backbones of each translator of the presented CDR are treated as being continuously constrained. The shape of each continuum joint (the PJ or the DJ) is hence represented by the shape of its imaginary central axis. It is inspired that each continuum joint is assumed to be an equivalent rod that coincides with the imaginary central axis, even though the deflection of a single rod is slightly different from that of a multi-backbone segment [30] (the experimental validation on the prototype shows good agreements in Sec. 4.4). The equivalent rod can be modeled with the Cosserat rod theory like that in Ref. [19].*Assumption*3.*Torsional Free*: Each equivalent rod is considered rigid in the torsional direction, because the external torsions exerted on the presented CDR may not be large for pick-and-place tasks.

*i*th translator’s PJ or DJ is a planar rod whose bending plane is the

*XY*plane of {

*i*1}. Thus, the kinetostatic model for each equivalent rod is regarded as a two-dimension form of the traditional Cosserat rod theory [16]. The kinetostatic model of the equivalent rod in the

*i*th translator is given in Eq. (11)

*s*(0 <

_{i}*s*<

_{i}*L*or

*L*+

*D*<

*s*<

_{i}*L*

_{trans}) is the centerline length of the equivalent rod in the

*i*th translator.

^{i}^{1}

**p**

*(*

_{i}*s*) ∈ ℝ

_{i}^{2}and

^{i}^{1}

**R**

*(*

_{i}*s*) ∈

_{i}*SO*(2) are the position vector in the XY plane of {

*i*1} and the rotation matrix about $z^$

_{i}_{1}, respectively, as a function of

*s*.

_{i}

^{i}^{1}

**n**

*(*

_{i}*s*) ∈ ℝ

_{i}^{2}and

*m*(

_{i}*s*) ∈ ℝ are the internal force in

_{i}*XY*plane of {

*i*1} and the internal moment along $z^$

_{i}_{1}, respectively, as a function of

*s*.

_{i}

^{i}^{1}

**v**

*(*

_{i}*s*) ∈ ℝ

_{i}^{2}and

*u*(

_{i}*s*) ∈ ℝ are the linear rate of

_{i}

^{i}^{1}

**p**

*(*

_{i}*s*) and the angular rate of

_{i}

^{i}^{1}

**R**

*(*

_{i}*s*), respectively.

_{i}

^{i}^{1}

**f**

*(*

_{i}*s*) ∈ ℝ

_{i}^{2}and

*l*(

_{i}*s*) ∈ ℝ are the external distributive force and moment exerted on the equivalent rod, as a function of

_{i}*s*.

_{i}**v**

**(*

_{i}*s*) = [10]

_{i}*and*

^{T}*u**(

_{i}*s*) = 0 are planar linear and angular rate, respectively, when the equivalent rod in the

_{i}*i*th translator is unstressed.

**K**

_{se}_{,i}and

*K*

_{bt}_{,i}=

*k*are two stiffness parameters related to shear/extension and bending deflections of the backbone, respectively.

_{i}IE*k*is the number of the backbone of the

_{i}*i*th translator.

*E*and

*I*are Young’s modulus and the cross-section second area moment of the backbone, respectively.

^{i}^{1}

**G**

*= [−*

_{m}*G*0]

_{m}*is the gravitational force of the translator. For simplification,*

^{T}

^{i}^{1}

**G**

*is assumed at the center of the multi-lumen tube.*

_{m}

^{i}^{1}

**p**

*∈ ℝ*

_{m}^{2}is the position of the center of multi-lumen tube with respect to the coordinate {

*i*1} in the

*XY*plane.

### 3.2 Boundary Conditions.

Equation (11) is integrated numerically while satisfying the following boundary conditions, considering the pose relations and the static equilibrium of three translators in the moving platform.

*Assumption*1,

^{i}^{1}

**R**

_{i}_{2}is equal to an identity matrix

**I**

_{2 × 2}. Thus, the pose boundary conditions for the

*i*th translator are set as follows:

^{i}^{1}

**R**

*)]*

_{i}^{˅}is the rotation angle of the rotation matrix

^{i}^{1}

**R**

*∈*

_{i}*SO*(2).

^{O}**p**

*is related to the position of the CDR’s moving platform with*

_{ide}

^{O}**p**

*=*

_{ide}

^{O}**p**

*+*

_{P}**r**

*.*

_{i}

^{O}**R**

_{i}_{1}in Eq. (16) is calculated using the actuation values

**q**= [

*q*

_{1}

*q*

_{2}

*q*

_{3}]

*via Eq. (18), using the expression of*

^{T}

^{O}**p**

*from Eq. (5)*

_{ipe}

^{O}**F**

*,*

_{e}

^{O}**l**

*∈ ℝ*

_{e}^{3}are the external force and moment applied at the moving platform in {

*O*}, respectively.

### 3.3 Numerical Solution Implementation.

The kinetostatic model for the CDR is solved by integrating the ordinary differential equations in Eq. (11) while satisfying the boundary conditions from Eqs. (16), (17), (19), and (20). The kinetostatic model includes the inverse and forward kinetostatic problems, which are both solved using a shooting method similar to the methods in Refs. [16,27].

^{O}**p**

*and*

_{P}**F**

*, respectively, while the actuation values*

_{e}**q**are unknown, is written in Eq. (21)

**ɛ**

_{IK}is the calculated residual error vector using the function

**b**

_{IK}(·) that is composed of Eqs. (16), (17), (19), and (20), and

**g**

_{IK}is the guess vector for the inverse kinetostatic model.

**g**

_{IK}is initially given and Eq. (11) for three translators are numerically integrated. The integral process is divided into two parts: integrate the equivalent rod of the PJ from

*s*= 0 to

_{i}*s*=

_{i}*L*and the equivalent rod of the DJ from

*s*=

_{i}*L*+

*D*to

*s*=

_{i}*L*

_{trans}. Then, the constraint equations (16), (17), (19), and (20) are applied to obtain the residual error vector

**ε**

_{IK}. Finally, the guess vector

**g**

_{IK}is updated iteratively using Eq. (22), until the norm of

*ɛ*_{IK}is less than a threshold (10

^{−4}in this paper)

*k*and

*λ*are the iteration index and damping factor, respectively, according to the Levenberg–Marquardt method.

**J**

_{IK,k}is the Jacobian matrix which is calculated numerically by taking a finite difference approach for the residual error vector

**ɛ**

_{IK,k}.

**q**and moving platform’s external force are given as [

*q*

_{1}

*q*

_{2}

*q*

_{3}]

*and*

^{T}**F**

*, respectively, while the moving platform’s position*

_{e}

^{O}**p**

*is unknown. This problem is formulated as follows:*

_{P}**ɛ**

_{FK}is the calculated residual error vector using the function

**b**

_{FK}(·) derived by Eqs. (16), (17), (19), and (20) and

**g**

_{FK}is the guess vector for the forward kinetostatic model. Equation (23) is solved similarly as the inverse kinetostatic model.

## 4 Experimental Characterizations

In this section, a series of experiments are conducted on the improved version of the CDR. The bending capability test is reported in Sec. 4.1. And the positioning accuracy is experimentally validated with and without loads in Sec. 4.2. Furthermore, the lateral force test and the bending moment test are demonstrated in Secs. 4.3 and 4.4, respectively. Lastly, a velocity evaluation for the pick-and-place task and a series of pick-and-place experiments are conducted in Sec. 4.5.

Software algorithms for the proposed model are developed and run on a PC (16 GB RAM and an 8-core Intel i7 10870H CPU @ 2.2 GHz) in c++ environment. The actual positions of the CDR’s moving platform are measured using an optical tracker (Micron Tracker Hx40, Claron Technology Inc.), referring to Fig. 6.

### 4.1 Bending Capability Test.

To illustrate the large bending capability of the CDR’s continuum joints, the CDR is actuated to two configurations whose actuation values are [190 mm 490 mm 490 mm]* ^{T}* (config 1), and [190 mm 490 mm 425 mm]

*(config 2). The calculated shapes of the CDR using the forward kinetostatic model from Eq. (23) and actual shapes are shown in Fig. 7. The maximal bending angles for the continuum joints are 94.11 deg and 89.98 deg for the two configurations.*

^{T}The result shows that the maximum bending angle of the continuum joints for the CDR is over 90 deg, which realizes and validates the requirement of the improved design for fully utilizing the bending angle ranges of the continuum joints.

### 4.2 Positioning Accuracy Without/With Load.

To demonstrate the proposed kinetostatic model’s accuracy, the CDR’s moving platform was commanded to move to 100 target positions without load and with a fixed 1000-gram load, as shown Figs. 8(a) and 9(a), respectively. The actuation values for the two loading conditions are calculated using the inverse kinetostatic model from Eq. (21) with the known external load. The 100 target positions are evenly distributed in four circles whose diameters are 150 mm, 150 mm, 300 mm, and 300 mm for the circles C1, C3, C2, and C4, respectively, with 15-deg interval (the first and the last positions have coincided). The centers of C1 and C2 are located at [0 0–175 mm]* ^{T}* while the ones of C3 and C4 are at [0 0–225 mm]

*, expressed in {*

^{T}*O*}. For comparison, the CDR was commanded to move to the same 100 target positions with actuation values calculated by the constant curvature assumption [22].

The positioning error curves without load and with a 1000-gram load are depicted in Figs. 8(b)–8(e) and 9(b)–9(e), respectively, while the average positioning errors for the four circular trajectories are listed in Table 2. The average positioning errors of all circles without load and with the 1000-gram load are 1.53 mm and 1.93 mm for the proposed kinetostatic model, compared with 2.75 mm and 4.43 mm for the constant curvature assumption [22] after tedious motion calibration and actuation compensation. Some positioning errors are analyzed as follows.

Positioning errors without load | Positioning errors with a 1000-gram load | |||
---|---|---|---|---|

Average position errors with the proposed model | Average position errors with the constant curvature assumption model | Average position errors with the proposed model | Average position errors with the constant curvature assumption model | |

C1 | 0.80 mm | 1.04 mm | 0.75 mm | 1.33 mm |

C2 | 1.29 mm | 1.62 mm | 1.55 mm | 2.64 mm |

C3 | 1.70 mm | 3.25 mm | 2.41 mm | 5.28 mm |

C4 | 2.39 mm | 5.09 mm | 3.08 mm | 8.48 mm |

Positioning errors without load | Positioning errors with a 1000-gram load | |||
---|---|---|---|---|

Average position errors with the proposed model | Average position errors with the constant curvature assumption model | Average position errors with the proposed model | Average position errors with the constant curvature assumption model | |

C1 | 0.80 mm | 1.04 mm | 0.75 mm | 1.33 mm |

C2 | 1.29 mm | 1.62 mm | 1.55 mm | 2.64 mm |

C3 | 1.70 mm | 3.25 mm | 2.41 mm | 5.28 mm |

C4 | 2.39 mm | 5.09 mm | 3.08 mm | 8.48 mm |

When no load is applied, the proposed kinetostatic model performs better than the constant curvature assumption model, because the CDR’s PJs and DJs diverse from the assumed constant curvature shape due to the gravities of translators and moving platform. Such as in Fig. 8, the positioning errors of the constant curvature assumption are relatively large at the point index of 5, 12, and 20 on the C2 and C4 circles. The reason is that the CDR’s PJs and DJs at these positions were bent with relatively large bending angles and the non-circular shapes similar in Fig. 1(b) appeared.

When a substantial external load is applied, the proposed kinetostatic model performs much better than the constant curvature assumption in the positioning accuracy. For example, in Figs. 9(b)–9(e), the positioning errors of the constant curvature assumption along the *z*-axis are almost minus due to the deflection in the loading direction. But these positioning errors caused by the loading effects can be reduced through the proposed kinetostatic model, as the external loads are modeled in Eq. (20). Nevertheless, some positioning errors still existed in the proposed kinetostatic model, as the PJs and the DJs of the CDR are not strengthened with some light-weight twist-resistant elements. But the torsion-free (*Assumption* 3 in Sec. 3) is assumed for the derivation of the proposed kinetostatic model.

### 4.3 Motion Test Under Lateral Loads.

To further evaluate the proposed kinetostatic model under lateral external loads, eight points evenly arranged in the C2 trajectory are selected as the targets of the moving platform. As shown in Fig. 10(a), the lateral forces along $x^$_{P} and $y^$_{P} were applied on the CDR’s moving platform using the different weights and a fixed pulley. The actuation values are calculated using the inverse kinetostatic model from Eq. (21) with the known external loads.

The positioning error curves with different lateral forces are depicted in Figs. 10(c) and 10(d), respectively. The average positioning errors with different lateral forces (0-gram, 100-gram, 200-gram, 500-gram, and 1000-gram) along $x^$_{P} are 1.58 mm, 1.92 mm, 1.93 mm, 2.71 mm, and 5.41 mm, respectively, while those along $y^$_{P} are 1.58 mm, 1.41 mm, 1.52 mm, 2.21 mm and 5.50 mm, respectively. The results show that when the average positioning errors are relatively small when the lateral force is less than 500 g. However, when a 1000-g lateral force was applied on the moving platform, the positioning errors are relatively large, due to the lack of twist-resistant structures on the PJs and the DJs of the CDR.

### 4.4 Motion Test Under Force and Moment.

To test the accuracy of the proposed kinetostatic model under the external moment, 25 points evenly arranged in the C2 trajectory are selected as the targets of the moving platform. As shown in Fig. 11(c), the external moment was applied using an axially loaded weight with a distance of *d*_{m} = 30 mm. Thus, both a vertical 10 N force and a 0.3 Nm moment were applied on the CDR’s moving platform. Then, the actuation values for the CDR are calculated using the inverse kinetostatic model from Eq. (21) with the known external force and moment.

The positioning and orientation error curves of the CDR’s moving platform are depicted in Figs. 11(b) and 11(d), respectively. The average positioning error for a 1000-g force and a 0.3 Nm bending moment is 2.39 mm, compared with 1.55 mm for a 1000-gram load and without moments (from Sec. 4.2). The two errors are close so it is considered that the positioning error with a 1000-g load and an 0.3 Nm bending moment exerted on the moving platform is reasonable. The orientation errors are calculated by the angle between $z^$_{P} and $z^$_{O}. The mean orientation errors with/without 0.3 Nm bending moment based on a 1000-g load are 2.28 deg and 2.50 deg, respectively. The result shows the reasonability of *Assumptions* 1 and 2, in which the bending angles of the PJ and the DJ as the equivalent rods are identical and in the same plane.

### 4.5 Pick-and-Place Experiments.

Furthermore, four pick-and-place tasks with objects of different weights and sizes were conducted to further validate the re-optimized CDR and the proposed kinetostatic model, as shown in Fig. 12. Different target positions of the moving platform were planned, and the actuation values were calculated using the inverse kinetostatic model of Eq. (21). And the speed used in the pick-and-place experiments were 0.4 m/s, 0.4 m/s, 0.4 m/s and 0.1 m/s with the 4-g load, 46-g load, 117-g load, and 1000-g load, respectively.

The results show that the re-optimized CDR can complete medium-speed and medium-load pick-and-place tasks. And the CDR brings an alternative for pick-and-place application.

## 5 Conclusion and Future Work

To enlarge the workspace of the initial design [22], the CDR’s structural parameters are re-optimized to fully utilize the bending angle ranges of the continuum joints, taking into considerations several possible structural interferences. After optimization, the maximum bending angle of the PJ or the DJ is 94.59 deg and the volume of the CDR’s workspace increases to 9.47 × 10^{7} mm^{3}, compared with 86.61 deg and 6.57 × 10^{7} mm^{3} for the initial design. Furthermore, a kinetostatic model for the CDR is derived based on the Cosserat rod theory to reduce the positioning errors whether with loads or not. Based on the proposed kinetostatic model, the experimental results indicate that the average positioning errors of four circle trajectories without load and with 1000-gram load are reduced to 1.53 mm and 1.93 mm, compared with 2.75 mm, and 4.43 mm for the constant curvature assumption model. Lastly, the re-optimized CDR and the proposed kinetostatic model were further validated through a series of pick-and-place tasks.

The future work mainly focuses on two aspects: (i) design a ight-weight twist-resistant element to avoid the undesired torsion motion of the CDR’s continuum joints, (ii) integrate a gripper at the CDR’s moving platform for more versatile pick-and-place tasks.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The data sets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

*i*=index of the CDR’s translator,

*i*= 1, 2, 3*s*=_{i}the centerline length along the

*i*th translator*L*_{trans}=total length of each translator. In this paper,

*L*_{trans}= 2*L*+*D*= 500 mm,*q**q*,_{i}*q*_{i,}_{min},*q*_{i,}_{max},*q*_{range}=**q**= [*q*_{1},*q*_{2},*q*_{3}].^{T}*q*is the actuation value of the_{i}*i*th translator along the axis of the*i*^{th}guideway.*q*= 0 when the origin of {_{i}*O*} coincides with that of {*ipe*}.*q*_{i,}_{min}and*q*_{i,}_{max}are the minimum and maximum value of*q*._{i}*q*_{range}=*q*_{i,}_{max}–*q*_{i,}_{min}*D*,_{i}*D*=length of the multi-lumen tube in the

*i*th translator, where*D*=*D*_{1}=*D*_{2}=*D*_{3}is assumed*L*,_{i}*L*=length of the PJ and the DJ in the

*i*th translator, where*L = L*_{1}=*L*_{2}=*L*_{3}is assumed*α*,_{i}*α*=inclination angle of the

*i*th guideway, where*α = α*_{1}=*α*_{2}=*α*_{3}is assumed*β*=_{i}division angle of the

*i*th kinematic chain, the angle between $x^$and projection of the_{O}*i*th guideway’s axis on the*XY*plane of {*O*}.*β*= (_{i}*i*−1)*π*/3*δ*=_{i}rotation angle from $y^$

_{i}_{1}to $x^$about $z^$_{ipe}$z^$_{ipe}._{ipe}*δ*∈ [−180 deg, 180 deg]_{i}*θ*,_{i}*θ*_{i,}_{max}=bending angle and the maximal bending angle of the

*i*th translator’s PJ and DJ*ϕ*=_{i}rotation angle from $x^$

to_{P}**r**about $z^$_{i}$z^$_{P}._{ipe}*ϕ*= (_{i}*i*−1)*π*/3