Design and modeling of continuum robot based on virtual-center of motion mechanism

doi:10.1007/s11465-022-0739-6

Front. Mech. Eng.

2023, Vol. 18

Issue (2) : 23 https://doi.org/10.1007/s11465-022-0739-6

RESEARCH ARTICLE

Design and modeling of continuum robot based on virtual-center of motion mechanism

Guoxin LI¹, Jingjun YU¹, Yichao TANG², Jie PAN¹, Shengge CAO¹, Xu PEI¹(

)

¹. School of Mechanical Engineering and Automation, Beihang University, Beijing 100191, China
². Beijing Special Engineering Design and Research Institute, Beijing 100143, China

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Abstract

Continuum robot has attracted extensive attention since its emergence. It has multi-degree of freedom and high compliance, which give it significant advantages when traveling and operating in narrow spaces. The flexural virtual-center of motion (VCM) mechanism can be machined integrally, and this way eliminates the assembly between joints. Thus, it is well suited for use as a continuum robot joint. Therefore, a design method for continuum robots based on the VCM mechanism is proposed in this study. First, a novel VCM mechanism is formed using a double leaf-type isosceles-trapezoidal flexural pivot (D-LITFP), which is composed of a series of superimposed LITFPs, to enlarge its stroke. Then, the pseudo-rigid body (PRB) model of the leaf is extended to the VCM mechanism, and the stiffness and stroke of the D-LITFP are modeled. Second, the VCM mechanism is combined to form a flexural joint suitable for the continuum robot. Finally, experiments and simulations are used to validate the accuracy and validity of the PRB model by analyzing the performance (stiffness and stroke) of the VCM mechanism. Furthermore, the motion performance of the designed continuum robot is evaluated. Results show that the maximum stroke of the VCM mechanism is approximately 14.2°, the axial compressive strength is approximately 1915 N/mm, and the repeatable positioning accuracies of the continuum robot is approximately ±1.47° (bending angle) and ±2.46° (bending direction).

Keywords VCM mechanism continuum robot flexural joint pseudo-rigid body model cable-driven

Corresponding Author(s): Xu PEI

Issue Date: 12 May 2023

Cite this article:

Guoxin LI,Jingjun YU,Yichao TANG, et al. Design and modeling of continuum robot based on virtual-center of motion mechanism[J]. Front. Mech. Eng., 2023, 18(2): 23.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-022-0739-6
https://academic.hep.com.cn/fme/EN/Y2023/V18/I2/23

Fig.1 Design of the virtual-center of motion mechanism: (a) structure of the virtual-center of motion mechanism and (b) structural parameters of the double leaf-type isosceles-trapezoidal flexural pivot.

Fig.2 Pseudo-rigid body model of a leaf-type isosceles-trapezoidal flexural pivot (LITFP): (a) analytic model of an LITFP, (b) equivalent four-bar model of an LITFP, (c) force analysis of the link DC, (d) bending angle at point D or C, and (e) equivalent pin-joint model of an LITFP.

Fig.3 Pseudo-rigid body model of the double leaf-type isosceles-trapezoidal flexural pivot (D-LITFP): (a) structure design of the D-LITFP and (b) equivalent pseudo-rigid body model of the D-LITFP.

Tab.1 Characteristic parameters of the two LITFPs

Tab.2 Characteristic parameters of the VCM mechanism

Fig.4 Influence of each parameter change on the stiffness: (a) change in stiffness K with the width b and thickness t, (b) change in stiffness K with height H and h_f, and (c) change in stiffness K with the included angle φ of the leaf.

Symbol	Description
$l single (i)$	Driving cable length in half joint
$Δ l single (i)$	Difference in driving cable length in half joint
$l joint (i)$	Driving cable length in single joint
$Δ l joint (i)$	Difference in driving cable length in single joint
$l segment (i)$	Driving cable length in a single segment
$Δ l segment (i)$	Difference in driving cable length in a single segment
$l total (i)$	Driving cable length of the whole continuum robot
$Δ l total (i)$	Difference in driving cable length of the whole continuum robot
$φ$	Bending angle of the half joint
$θ j o i n t$ , $η j o i n t$	Bending angle and bending direction of the single joint
$θ s e g m e n t$ , $η s e g m e n t$	Bending angle and bending direction of the single segment
$T s e g m e n t$	Pose transformation matrix of the single segment
$T t o t a l$	End pose transformation matrix of the whole continuum robot

Tab.3 Symbol description table

Fig.5 Mappings among driving space, configuration space, and task space.

Fig.6 Design of the flexural joint: (a) combination mode of half joint and (b) coordinate transformation of half joint.

Fig.7 Half joint: (a) analytical model of cable length difference for half joint and (b) distribution, and a serial number of driving cables (uniform distribution). Each segment is driven by three evenly distributed driving cables, of which Nos. 1, 4, and 7 driving cables are the first group, Nos. 2, 5, and 8 driving cables are the second group, and Nos. 3, 6, and 9 driving cables are the third group.

Fig.8 Schematic of single-segment bending: (a) bending plane and (b) coordinate transformation.

Fig.9 Axial stiffness simulation of virtual-center of motion mechanism: (a) equivalent stress and (b) axial deformation.

Fig.10 Finite element analysis (FEA): (a) FEA and pseudo-rigid body (PRB) values of stiffness for LITFP 1, (b) FEA and PRB values of stiffness for LITFP 2, (c) relative errors of stiffness for LITFPs 1 and 2, (d) FEA and PRB values of maximum stress for LITFP 1, (e) FEA and PRB values of maximum stress for LITFP 2, (f) relative errors of maximum stress for LITFPs 1 and 2, (g) FEA and PRB values of stiffness for virtual-center of motion (VCM) mechanism, (h) FEA and PRB values of maximum stress for VCM mechanism, and (i) relative errors of stiffness and maximum stress for VCM mechanism. LITFP: leaf-type isosceles-trapezoidal flexural pivot.

Fig.11 Stiffness experiment of virtual-center of motion mechanism: (a) experimental device and (b) experimental scene.

Tab.4 Comparison of the performance of the VCM mechanism for two different materials.

Fig.12 Trend of experimental stiffness and theoretical calculation stiffness of samples (t = 1.0, 1.2, and 1.4 mm).

Fig.13 Workspaces of the continuum robot.

Fig.14 Motion performance of the continuum robot: (a) prototype and control system of continuum robot, (b) C-shape, (c) S-shape, and (d) uniform circular motion (

0 ∘ − 180 ∘

Tab.5 Repeatable positioning accuracy experiment arrangement

Fig.15 Repeatable positioning accuracy experiment: repeatable positioning accuracy of (a) bending angle and (b) bending direction; deviation angle of (c) bending angle and (d) bending direction.

Abbreviations
Al alloy	Aluminum alloy
CLITFP	Compressed leaf-type isosceles-trapezoidal flexural pivot
D-LITFP	Double leaf-type isosceles-trapezoidal flexural pivot
FEA	Finite element analysis
ICR	Instantaneous center of rotation
IE	Intermediate element
LITFP	Leaf-type isosceles-trapezoidal flexural pivot
ME	Movement element
PLA	Polylactic acid
PRB	Pseudo-rigid body
S	Stand
TLITFP	Tensioned leaf-type isosceles-trapezoidal flexural pivot
VCM	Virtual-center of motion
Variables
a₁, a₂	X-coordinate of the end points B and C of link BC
b	Width of the leaf
$C C ′ ¯, D D ′ ¯$	Displacements of points C and D, respectively
E	Elastic modulus
I, I_i	Moments of inertia of the leaf and LITFP i, respectively
F	Force
$F C x$ , $F C y$	Component forces at point C on the X- and Y-axis, respectively
$F D x$ , $F D y$	Component forces at point D on the X- and Y-axis, respectively
$F N C$ , $F N D$	Axial forces applied to link DC on points C and D, respectively
F_RC, F_RD	Radial forces exerted by link BC and AD on points C and D of link DC, respectively
h_f	Height of the lower plane of LITFP from the ICR
h_fi	Height of the lower plane of LITFP i from the ICR, i = 1, 2
H	Height of the upper plane of LITFP from the ICR
H_i	Height of the upper plane of LITFP i from the ICR, i = 1, 2
K	Bending stiffness of LITFP
K_BC, K_AD	Bending stiffness of links BC and AD, respectively
K_d	Bending stiffness of the D-LITFP
K_i	Bending stiffness of the LITFP i, i = 1, 2
K_V	Bending stiffness of the VCM mechanism
$l j o i n t (i)$	Driving cable length in single joint
l_r	Length of the rigid links $A ′ D$ and $B ′ C$
$l s e g m e n t (i)$	Driving cable length in a single segment
$l s i n g l e (i)$	Driving cable length in half joint
$l t o t a l (i)$	Driving cable length of the whole continuum robot
M	A pure bending moment
$M max$	Maximum bending moment which LITFP can bear
n	Position coefficient of ICR
n_i	Position coefficient of the ICR of the LITFP i
r	Radius of the circle where the driving cable is located
$r f$	Result of FEA
$r p$	Calculation result of the PRB model
R	Bending radius
$R b e n d$	Bending radius of the segment
$R Y$	Rotation matrix around the Y-axis
$R Z$	Rotation matrix around the Z-axis
$04 R$ , $04 P$	Rotation matrix and displacement vector from ${O 4}$ to ${O 0}$ , respectively
S_y	Tensile yield strength
t	Thickness of the leaf
t_i	Thickness of the leaf of LITFP i
$T s e g m e n t$	Pose transformation matrix of the single segment
$T t o t a l$	End pose transformation matrix of the whole continuum robot
$01 T$	Coordinate transformation matrix from ${O 1}$ to ${O 0}$
$02 T$	Coordinate transformation matrix from ${O 2}$ to ${O 0}$
$12 T$	Coordinate transformation matrix from ${O 2}$ to ${O 1}$
$α 1$ , $α 2$	Bending angles of link BC and AD under the action of bending moment M
$σ 1 max$ , $σ 2 max$	Maximum stress values corresponding to rotation angles of LITFPs 1 and 2, respectively
σ_dmax, σ_max	Maximum stress of the D-LITFP and LITFP, respectively
$φ$	Bending angle of the half joint
φ_i	Half of the angle between the two leaves of the LITFP i, i = 1, 2
$φ X$ , $φ Y$	X- and Y-axis tilt angles, respectively
θ	Rotation angle (stroke) of the VCM mechanism
$θ d$	Rotation angle of the whole D-LITFP
$θ d max$	Maximum bending angle of the whole D-LITFP
θ_i, θ_imax	(Maximum) Bending angle of the LITFP i, i = 1, 2
$θ j o i n t$ , $θ s e g m e n t$	Bending angles of the single joint and single segment, respectively
θ_max	Maximum bending angle of the LITFP
$η$	Bending direction
$η j o i n t$ , $η s e g m e n t$	Bending directions of the single joint and single segment, respectively
$ε$	Relative error of PRB model with respect to FEA
$Δ l$	Displacement of the force sensor
$Δ l (i)$	Cable length difference
$Δ l j o i n t (i)$	Difference in driving cable length in a single joint
$Δ l s e g m e n t (i)$	Difference in driving cable length in a single segment
$Δ l s i n g l e (i)$	Difference in driving cable length in half joint
$Δ l t o t a l (i)$	Difference in driving cable length of the whole continuum robot

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