A modular cable-driven humanoid arm with anti-parallelogram mechanisms and Bowden cables

doi:10.1007/s11465-022-0722-2

Frontiers of Mechanical Engineering

2023, Vol. 18

Issue (1): 6 https://doi.org/10.1007/s11465-022-0722-2

本期目录

A modular cable-driven humanoid arm with anti-parallelogram mechanisms and Bowden cables

Bin WANG, Tao ZHANG(

), Jiazhen CHEN, Wang XU, Hongyu WEI, Yaowei SONG, Yisheng GUAN(

)

School of Electro-mechanical Engineering, Guangdong University of Technology, Guangzhou 510006, China

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Abstract：

This paper proposes a novel modular cable-driven humanoid arm with anti-parallelogram mechanisms (APMs) and Bowden cables. The lightweight arm realizes the advantage of joint independence and the rational layout of the driving units on the base. First, this paper analyzes the kinematic performance of the APM and uses the rolling motion between two ellipses to approximate a pure-circular-rolling motion. Then, a novel type of one-degree-of-freedom (1-DOF) elbow joint is proposed based on this principle, which is also applied to design the 3-DOF wrist and shoulder joints. Next, Bowden cables are used to connect the joints and their driving units to obtain a modular cable-driven arm with excellent joint independence. After that, both the forward and inverse kinematics of the entire arm are analyzed. Last, a humanoid arm prototype was developed, and the assembly velocity, joint motion performance, joint stiffness, load carrying, typical humanoid arm movements, and repeatability were tested to verify the arm performance.

Key words： modular robotic arm anti-parallelogram mechanism Bowden cable humanoid arm lightweight joint design

收稿日期: 2022-02-17 出版日期: 2023-02-16

Corresponding Author(s): Tao ZHANG,Yisheng GUAN

引用本文:

. [J]. Frontiers of Mechanical Engineering, 2023, 18(1): 6.
Bin WANG, Tao ZHANG, Jiazhen CHEN, Wang XU, Hongyu WEI, Yaowei SONG, Yisheng GUAN. A modular cable-driven humanoid arm with anti-parallelogram mechanisms and Bowden cables. Front. Mech. Eng., 2023, 18(1): 6.

链接本文:

https://academic.hep.com.cn/fme/CN/10.1007/s11465-022-0722-2
https://academic.hep.com.cn/fme/CN/Y2023/V18/I1/6

Fig.1

Fig.2

Joint	$l p / m m$	$s p / m m$	$d 0 / m m$	Max deviation/mm	Min deviation/mm
Wrist	60.00	20	3.05	0.060	−0.052
Elbow	65.00	28	5.70	0.120	−0.112
Shoulder	85.75	30	4.84	0.092	−0.094

Tab.1

Fig.3

Fig.4

Fig.5

Joint	Pitch/(° )	Yaw/(° )	Roll/(° )
Wrist	$[− 90, 90]$	$[− 90, 90]$	$[− 720, 720]$
Elbow	$[− 90, 90]$	?	?
Shoulder	$[0, 180]$	$[− 90, 90]$	$[− 180, 180]$

Tab.2

Fig.6

Fig.7

Fig.8

Fig.9

Tab.3

Fig.10

Fig.11

Fig.12

Fig.13

Fig.14

Fig.15

Fig.16

Tab.4

Abbreviations
3D	Three-dimensional
APM	Anti-parallelogram mechanism
DOF	Degree-of-freedom
IMU	Inertial measurement unit
MCDH-Arm	Modular cable-driven humanoid arm
PCR	Pure-circular-rolling
PCRM	Pure-circular-rolling mechanism
SAPM	Spatial anti-parallelogram mechanism
Variables
$[A d s T t]$	Adjoint matrix of $s T t$
$c 0$	Circle centered of the desired deflection trajectory of the APM
$d (θ A)$	Deviation between ellipse and circle
$d 0$	Distance between $c 0$ and the x-axis
$h p$	Distance of the top and bottom links of parallogram when they are parallel
$i v$	Linear velocity reduction ratio of the APM
$i ω$	Angular velocity reduction ratio of the APM
$J s$	Jacobian matrix
$J s †$	Pseudoinverse of Jacobian $J s$
$k$ , $k e$ , $k s$	Center distances of the pulleys on the two antagonistic pulley groups of the APM, elbow, and shoulder, respectively
$l e$	Length of $O e O f$
$l h$	Length of $O h O t$
$l p$	Length of the two intersecting links of the APM
$l s$	Length of $O s O r$
$l u r$	Length of $O r O u$
$l w$	Length of $O w O h$
$Δ l$	Cable length change during the APM movement
$Δ l 1$	Connecting rod length of forearm
$Δ l 2$	Connecting rod length of upper arm
m	Number of the pulley groups
$M a x (v A e − v A c)$	Maximum linear velocity difference of point A on the moving platform between the APM and the PCRM
n	Number of driving cables on each side of the APM
$n e$ , $n s$ , $n w$	Numbers of cables on the pulley group of the elbow, shoulder, and wrist, respectively
$P c$	Desire intersection point of the two circles of the APM
$P e n d i$ , $P s t a r t i$	End points and start points coordinates of each trajectory, respectively
$P p$	Intersection point of the two sides of the parallogram
$s p t$	Translation vector of the transformation matrix $s T t$
$s p ˙ t$	Differential of the translation vector $s p t$
$Δ p s$	Pitch cable length changes of the shoulder
r	$r = Δ l / Δ l (n e k e) (n e k e) = sin ? θ A$ is used to simplify $sin ? θ A$ of elbow transform matrix $e T f$
p	$p = 1 − (Δ l / Δ l (n e k e) (n e k e)) 2$ is used to simplify $cos ? θ A$ of elbow transform matrix $e T f$
$r A$	Distance between points A and $O 2$
$r c$	Distance between $P c$ and $c 0$
$r m$	Motor-driven reel radius of the APM
$r p$	Distance between $P p$ and $c 0$
$r w$	Cable length change of the wrist reel
$R w$	Reel radius change of the wrist
$s R e$	Rotation transformation matrix of elbow fixed platform $O e x e y e z e$ relative to the base frame
$s R w$	Rotation transformation matrix of wrist fixed platform $O w x w y w z w$ relative to the base frame
$s p$	Focal distance of the ellipse
$s T d$	Transformation matrix of desired pose with respect to the base frame $O s x s y s z s$
$t T d (θ i)$	Transformation matrix of desired pose relative to the actual pose of the end effector $O t x t y t z t$
$u T e$	Transformation matrix between $O u x u y u z u$ and $O e x e y e z e$
$e T f$	Transformation matrix between $O f x f y f z f$ and $O e x e y e z e$
$w T h$	Transformation matrix between $O w x w y w z w$ and $O h x h y h z h$ , kinematic transformation matrix of the 2-DOF wrist SAPM
$s T r$	Transformation matrix between $O r x r y r z r$ and $O s x s y s z s$ , kinematics equation of the 2-DOF shoulder SAPM
$h T t$	Transformation matrix between $O h x h y h z h$ and $O t x t y t z t$
$s T t$	Transformation matrix between $O t x t y t z t$ and $O s x s y s z s$ , overall forward kinematics of the arm
$r T u$	Transformation matrix matrix between $O u x u y u z u$ and O_rx_ry_rz_r
$f T w$	Transformation matrix between $O f x f y f z f$ and $O w x w y w z w$
$v A$	Linear velocity of point A on the moving platform, it include linear velocity $v A c$ of PCRM (circle trajectory) and linear velocity $v A e$ of APM (ellipse trajectory)
$v A c$	Linear velocity of point A on the PCRM moving platform (circle trajectory)
$v A e$	Linear velocity of point A on the APM moving platform (ellipse trajectory)
$v l$	Cable linear velocity of the APM
$v O 2$	Linear velocity of O₂ rotating around O₁ of the APM
$v t$	Linear velocity of the end effector in the base frame
$V s$	Twist of the desired pose relative to the base frame
$V t$	Twist of the desired pose relative to the actual pose of the end effector
$[V t]$	Antisymmetric matrix of twist $V t$
$Δ w s$	Yaw cable length changes of the shoulder
$x p$	Abscissa of $P p$
$y p$	Ordinate of $P p$
α	Ratio of the maximum linear velocity difference to the maximum linear velocity of the PCRM
$θ A$	Angle at which the connecting link $O 1 O 2$ deviates from the initial position of the APM
$θ e$	Angle between link $O e O f$ and the horizontal direction of the elbow
$θ h$	Rotation angle of the wrist rotating mechanism, $θ h = r w / r w R w R w$
$θ i$	Humanoid arm joint angle of the ith iteration
$θ p$	Angle between the link $O s O r$ and its initial position of the shoulder
$θ u$	Rotation angle of the shoulder rotating body
$θ w$	Angle between the link $O w O h$ and its initial position of wrist
$λ$	$λ = Δ w s / Δ w s (n s k s) (n s k s)$ is used to simplify $sin ? φ s sin ? θ p$ of transform matrix $s T r$
$γ$	$γ = Δ p s / = Δ p s (n s k s) (n s k s)$ is used to simplify $cos ? φ s sin ? θ p$ of transform matrix $s T r$
$τ$	$τ 2 = 1 − λ 2 − γ 2$ is used to simplify $cos 2 θ p$ of transform matrix $s T r$
$φ s$	Yaw angle of the bending plane of the shouder
$φ w$	Yaw angle of the bending plane of the wrist
$ψ$	$ψ = sin ? φ w sin ? θ w$ is used to simplify $sin ? φ w sin ? θ w$ of transform matrix $s T w$
$σ$	$σ = cos ? φ w sin ? θ w$ is used to simplify $cos ? φ w sin ? θ w$ of transform matrix $s T w$
$η$	$η 2 = 1 − ψ 2 − σ 2$ is used to simplify $cos 2 θ w$ of transform matrix $s T w$
$ω 1$	Angular velocity of the connecting link $O 1 O 2$ rotating around $O 1$ of the APM
$ω 2$	Angular velocity of the moving platform rotating around $O 2$ of the APM
$ω f$	Angular velocity of the elbow in the base frame
$ω m$	Motor angular velocity
$ω t$	Angular velocity of the end effector in the base frame

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