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

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).

Key words： VCM mechanism; continuum robot; flexural joint; pseudo-rigid body model; cable-driven

Nomenclature

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
\overline{C{C}^{\prime }},\overline{D{D}^{\prime }}	Displacements of points C and D, respectively
E	Elastic modulus
I, Ii	Moments of inertia of the leaf and LITFP i, respectively
F	Force
F_{Cx}, F_{Cy}	Component forces at point C on the X- and Y-axis, respectively
F_{Dx}, F_{Dy}	Component forces at point D on the X- and Y-axis, respectively
F_{\mathrm{N}C}, F_{\mathrm{N}D}	Axial forces applied to link DC on points C and D, respectively
FRC, FRD	Radial forces exerted by link BC and AD on points C and D of link DC, respectively
hf	Height of the lower plane of LITFP from the ICR
hfi	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
Hi	Height of the upper plane of LITFP i from the ICR, i = 1, 2
K	Bending stiffness of LITFP
KBC, KAD	Bending stiffness of links BC and AD, respectively
Kd	Bending stiffness of the D-LITFP
Ki	Bending stiffness of the LITFP i, i = 1, 2
KV	Bending stiffness of the VCM mechanism
l_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}\left(i\right)	Driving cable length in single joint
lr	Length of the rigid links A^{\prime }D and B^{\prime }C
l_{\mathrm{s}\mathrm{e}\mathrm{g}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}}\left(i\right)	Driving cable length in a single segment
l_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}}\left(i\right)	Driving cable length in half joint
l_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}\left(i\right)	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
ni	Position coefficient of the ICR of the LITFP i
r	Radius of the circle where the driving cable is located
r_{\mathrm{f}}	Result of FEA
r_{\mathrm{p}}	Calculation result of the PRB model
R	Bending radius
R_{\mathrm{b}\mathrm{e}\mathrm{n}\mathrm{d}}	Bending radius of the segment
{\mathit{R}}_Y	Rotation matrix around the Y-axis
{\mathit{R}}_Z	Rotation matrix around the Z-axis
{}_0^4\mathit{R}, {}_0^4\mathit{P}	Rotation matrix and displacement vector from \left\{O_4\right\} to \left\{O_0\right\}, respectively
Sy	Tensile yield strength
t	Thickness of the leaf
ti	Thickness of the leaf of LITFP i
{\mathit{T}}_{\mathrm{s}\mathrm{e}\mathrm{g}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}}	Pose transformation matrix of the single segment
{\mathit{T}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}	End pose transformation matrix of the whole continuum robot
{}_0^1\mathit{T}	Coordinate transformation matrix from \left\{O_1\right\} to \left\{O_0\right\}
{}_0^2\mathit{T}	Coordinate transformation matrix from \left\{O_2\right\} to \left\{O_0\right\}
{}_1^2\mathit{T}	Coordinate transformation matrix from \left\{O_2\right\} to \left\{O_1\right\}
{\alpha }_1, {\alpha }_2	Bending angles of link BC and AD under the action of bending moment M
{\sigma }_{1max}, {\sigma }_{2max}	Maximum stress values corresponding to rotation angles of LITFPs 1 and 2, respectively
σdmax, σmax	Maximum stress of the D-LITFP and LITFP, respectively
\phi	Bending angle of the half joint
φi	Half of the angle between the two leaves of the LITFP i, i = 1, 2
{\phi }_X, {\phi }_Y	X- and Y-axis tilt angles, respectively
θ	Rotation angle (stroke) of the VCM mechanism
{\theta }_{\mathrm{d}}	Rotation angle of the whole D-LITFP
{\theta }_{\mathrm{d}max}	Maximum bending angle of the whole D-LITFP
θi, θimax	(Maximum) Bending angle of the LITFP i, i = 1, 2
{\theta }_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}, {\theta }_{\mathrm{s}\mathrm{e}\mathrm{g}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}}	Bending angles of the single joint and single segment, respectively
θmax	Maximum bending angle of the LITFP
\eta	Bending direction
{\eta }_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}, {\eta }_{\mathrm{s}\mathrm{e}\mathrm{g}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}}	Bending directions of the single joint and single segment, respectively
\epsilon	Relative error of PRB model with respect to FEA
\mathrm{\Delta }l	Displacement of the force sensor
\mathrm{\Delta }l\left(i\right)	Cable length difference
\mathrm{\Delta }{l}_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}\left(i\right)	Difference in driving cable length in a single joint
\mathrm{\Delta }{l}_{\mathrm{s}\mathrm{e}\mathrm{g}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}}\left(i\right)	Difference in driving cable length in a single segment
\mathrm{\Delta }{l}_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}}\left(i\right)	Difference in driving cable length in half joint
\mathrm{\Delta }{l}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}\left(i\right)	Difference in driving cable length of the whole continuum robot

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant No. U1813221) and the National Key R&D Program of China (Grant No. 2019YFB1311200).

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