Motion modeling of cable-driven continuum robots using vector form intrinsic finite element method

ABSTRACT

This paper presents the motion modeling of a cable-driven, single backbone continuum robot via the vector form intrinsic finite element (VFIFE) method. The VFIFE method is a solution framework based on vector mechanics. The algorithm describes a continuous body by using a finite number of particles instead of a mathematical function. We first approached the model by defining path elements and allocating particles (mass nodes) for the structure. We then established constitutive conditions as the generalized forces derived from both the external environment and internal deformed structural elements. We found that the dynamic equation of particles can be governed by Newton’s law of motion and solved by a general explicit time integration technique. Furthermore, we performed static modeling of the system by employing a dynamic relaxation algorithm with kinetic damping in the dynamic equations. Finally, from experiments, we validated the simulated static model of a single backbone continuum robot. It is shown that this method is capable of describing a continuum robot’s motion by solving for frame structures that undergo large deformation.

KEYWORDS:

• Continuum robot
• vector form intrinsic finite element
• dynamic relaxation
• kinetic damping

Nomenclature

$\mathit{d}$/$\dot{\mathit{d}}$/$\dot{\mathit{d}}$ displacement/velocity/acceleration vector of a particle

d_t displacement of particle at current time step in time integration

$d_{t+\mathrm{\Delta }t}$ displacement of particle at next time step in time integration

(${\mathbf{e}}_x^a$,${\mathbf{e}}_y^a$,${\mathbf{e}}_z^a$) local coordinate system at t=t_a

(${\mathbf{e}}_x^t$,${\mathbf{e}}_y^t$,${\mathbf{e}}_z^t$) local coordinate system at t=t

${\mathbf{f}}_{\mathrm{i}\mathrm{n}\mathrm{t}}^i$ internal force vector due to deformation in element

f_N normal contact force between cable and disk

f_fr induced friction force due to f_N

F_ext external force vector

F_g,i gravitational force at node i

I_j mass moment of inertia of particle j

${\hat{J}}_{ya}$, ${\hat{J}}_{za}$ area moment of inertia

${\hat{J}}_{xa}$ polar moment of inertia

l, l_a element lengths at time t and t_a

M mass matrix of a particle

${\mathbf{R}}_a^t$ rotation matrix from coordinate t=t_a to t=t

T_ext cable pulling force

T(t) cable pulling force in time function

T_i, T_i+1 cable pulling force in vector form

u¹, u² relative change of translational nodal displacement of nodes 1 and 2

x position vector of the particle

βⁱ relative change for nodal angle of node i

${\stackrel{\mathbf{ˆ}}{\beta }}^{id}$ deformation of the nodal angle of node i

${\beta }_{\mathrm{x}}^1$ component of rigid body rotation angle

${\hat{\mathrm{\Delta }}}_e$ axial stretch of the element

${\theta }_{at}$ component of rigid body rotation angle in ${\mathbf{e}}^{at}$ direction

${\gamma }_t$ rigid body rotation vector from time t_a to t

ψ_i angle between two cable tensions of a disk

$\mathrm{\Delta }t$ time step in time integration

${\mathbf{\Omega }}_a$ coordinate transformation matrix from global to local

Acknowledgments

The financial support of Ministry of Science and Technology of Taiwan (MOST 108-2221-E-002-132-) as well as National Taiwan University (NTU) to the authors is acknowledged.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Ministry of Science and Technology, Taiwan [108-2221-E-002-132]; National Taiwan University [108-2926-I-002-002-MY4].