Optimum design and preliminary experiments of a novel parallel end traction apparatus for upper-limb rehabilitation

Abstract

Robot-assisted technology has been increasingly employed in the therapy of post stroke patients to deliver high-quality treatment and alleviate therapists’ burden. This paper introduces a novel parallel end traction apparatus (PETA) to supplement equipment selection. Considering the appearance and performance of the PETA, two types of special five-bar linkage mechanisms are selected as the potential configurations of the actuation execution unit because of their compact arrangement and parallel structure. Kinematic analysis of each mechanism, i.e., position solutions and Jacobian matrix, is carried out. Subsequently, a comparative study between the two mechanisms is conducted. In the established source of nondimensional parameter synthesis, the singularity, maximum continuous workspace, and performance variation trends are analyzed. Based on the evaluation results, the final scheme with determined configuration and corresponding near-optimized nondimensional parameters is obtained. Then, a prototype is constructed. By adding a lockable translational degree of freedom in the vertical direction, the PETA can provide 2D planar exercise and 3D spatial exercise. Finally, a control system is developed for passive exercise mode based on the derived inverse position solution, and preliminary experiments are performed to verify the applicability of the PETA.

Graphical abstract

Keywords

parallel mechanism / upper-limb rehabilitation / singularity and workspace analyses / performance evaluation / optimum design

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Nomenclatures

Abbreviations
DOF	Degree of freedom
MCW	Maximum continuous workspace
PETA	Parallel end traction apparatus
ROM	Range of motion
RTTA	Rectangular target treatment area
SPTM	Special parallelogram type mechanism
SSTM	Special symmetrical type mechanism
Variables
Bi	Coordinate of point Bi
Di	DOFs permitted by joints
f1	Output force along X-axis
f2	Output force along Y-axis
f	Vector of output forces
F	DOFs of SSTM and SPTM
Gris	Global maximum force index
GηJ	Global motion isotropy index
GηS	Global structural stiffness index
h	Number of joints
JFoSPTM	Forward Jacobian matrix of SPTM
JFoSSTM	Forward Jacobian matrix of SSTM
JIoSPTM	Inverse Jacobian matrix of SPTM
JIoSSTM	Inverse Jacobian matrix of SSTM
JS	Stiffness Jacobian matrix
JV	Velocity Jacobian matrix
JVoSPTM	Velocity Jacobian matrix of SPTM
JVoSSTM	Velocity Jacobian matrix of SSTM
K	Scalar matrix representing the stiffness of the active joints
k1, k2	Stiffness of of joints A1 and A2, respectively
l1,l2	Nondimensional form of L1 andL2, respectively
lvlp	Length of long principal axis of velocity ellipsoid
lvsp	Length of short principal axis of velocity ellipsoid
L	Average value of L1 and L2
L1	Length of links AiBi of SSTM, lengths of links A1B1 and B2P of SPTM
L2	Length of links BiP of SSTM, lengths of links A2B2 and B1P of SPTM
n	Number of links
P	Coordinate of point P
\dot{\mathit{q}}	Vector of input velocities

ris	Local maximum force index, i.e., radius of inscribed circle contained in Tf
Tf	Generalized set of output forces of end effector
Tτ	Set of allowable torques of active joints
\dot{\mathit{u}}	Vector of output velocities
v	Number of parallel redundant constraints
w	MCW of mechanism
XP	Coordinate of point P in X-axis direction
{\dot{X}}_P	Output velocity in X-axis direction
YP	Coordinate of point P in Y-axis direction
{\dot{Y}}_P	Output velocity in Y-axis direction
ηJ	Local motion isotropy index, i.e., inverse value of condition number of velocity ellipsoid
ηS	Local structural stiffness index, i.e., maximum micro deformation of end effector
θ1	Input angle of joint A1
{\dot{\theta }}_1	Input angular velocity of joint A1
θ2	Input angle of joint A2
{\dot{\theta }}_2	Input angular velocity of joint A2
λ	Number of common constraints
λi	Eigenvalues of matrix JSTJS
τ	Vector of driving torques
τ1, τ2	Driving torque of joints A1 and A2, respectively
τ1max, τ2max	Maximum torque applied by actuator on joints A1 and A2, respectively
Δq	Vector of virtual angular displacements associated with active joints
Δu	Vector of virtual deformations of end effector
ΔXP, ΔYP	Virtual deformation of end effector in X- and Y-axis directions, respectively
Δθ1, Δθ2	Virtual angular displacement associated with joint A1 and A2, respectively

Acknowledgements

The authors thank Henan Huibo Medical Co., Ltd. for several useful suggestions on this apparatus. Additionally, this research is funded in part by Beijing Natural Science Foundation (Grant No. 3204036), in part by the National Key R&D Program of China (Grant Nos. 2018YFB1307004 and 2020YFC2004200), in part by the National Natural Science Foundation of China (Grant No. 61903011), and in part by the General Program of Science and Technology Development Project of the Beijing Municipal Education Commission (Grant No. KM202010005021).

Electronic Supplementary Materials

The supplementary materials can be found in the online version of this article at https://doi.org/10.1007/s11465-021-0651-5 and are accessible to authorized users.

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