Design of a novel side-mounted leg mechanism with high flexibility for a multi-mission quadruped earth rover BJTUBOT

doi:10.1007/s11465-022-0740-0

Front. Mech. Eng.

2023, Vol. 18

Issue (2) : 24 https://doi.org/10.1007/s11465-022-0740-0

RESEARCH ARTICLE

Design of a novel side-mounted leg mechanism with high flexibility for a multi-mission quadruped earth rover BJTUBOT

Yifan WU, Sheng GUO(

), Luquan LI, Lianzheng NIU, Xiao LI

School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, China

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Abstract

Earth rover is a class of emerging wheeled-leg robots for nature exploration. At present, few methods for these robots’ leg design utilize a side-mounted spatial parallel mechanism. Thus, this paper presents a complete design process of a novel 5-degree-of-freedom (5-DOF) hybrid leg mechanism for our quadruped earth rover BJTUBOT. First, a general approach is proposed for constructing the novel leg mechanism. Subsequently, by evaluating the basic locomotion task (LT) of the rover based on screw theory, we determine the desired motion characteristic of the side-mounted leg and carry out its two feasible configurations. With regard to the synthesis method of the parallel mechanism, a family of concise hybrid leg mechanisms using the 6-DOF limbs and an L_1F1C limb (which can provide a constraint force and a couple) is designed. In verifying the motion characteristics of this kind of leg, we select a typical (3-UPRU&RRRR)&R mechanism and then analyze its kinematic model, singularities, velocity mapping, workspace, dexterity, statics, and kinetostatic performance. Furthermore, the virtual quadruped rover equipped with this innovative leg mechanism is built. Various basic and specific LTs of the rover are demonstrated by simulation, which indicates that the flexibility of the legs can help the rover achieve multitasking.

Keywords design synthesis parallel mechanism hybrid leg mechanism screw theory quadruped robot

Corresponding Author(s): Sheng GUO

Just Accepted Date: 09 October 2022 Issue Date: 05 June 2023

Cite this article:

Yifan WU,Sheng GUO,Luquan LI, et al. Design of a novel side-mounted leg mechanism with high flexibility for a multi-mission quadruped earth rover BJTUBOT[J]. Front. Mech. Eng., 2023, 18(2): 24.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-022-0740-0
https://academic.hep.com.cn/fme/EN/Y2023/V18/I2/24

Fig.1 Concept of constructing a side-mounted novel leg mechanism for our earth rover. DOF: degree-of-freedom, MP: moving platform.

Fig.2 d-dimensional wrench system on the output link of the side-mounted leg mechanism.

Fig.3 Motion forms only use rotation degree of freedoms: (a) deployable function, (b) omnidirectional motion, (c) Ackerman steering, and (d) pivot steering.

Fig.4 Motion forms only use translation degree of freedoms: (a) active suspension function and (b) walking motion.

Fig.5 Special functions combine different motion forms: (a) adaptive walking, (b) crawling, and (c) jumping.

Fig.6 Wrench of the two feasible configurations of the side-mounted leg mechanism: (a) 5-DOF side-mounted parallel leg mechanism and (b) hybrid leg mechanism using a 4-DOF side-mounted parallel mechanism and a revolute steering joint. DOF: degree-of-freedom.

Fig.7 Some configurations of 3R2T 5-degree-of-freedom parallel mechanisms using a symmetrical structure with F-limbs: (a) 5-5R(R₂R₂R₂R₁R₁) and (b) 5-3R2P(PPR₂R₁R₁). P: prismatic joint, R: revolute joint, R₁: type-1 revolute joint, R₂: type-2 revolute joint.

Fig.8 Some configurations of 2R2T 4-degree-of-freedom parallel mechanisms using a symmetrical structure with F-limbs: (a) 4-5R(R₂R₂R₂R₁R₁) and (b) 4-4R1P(PR₂R₂R₁R₁). P: prismatic joint, R: revolute joint, R₁: type-1 revolute joint, R₂: type-2 revolute joint.

Fig.9 Two examples of 3R2T 5-DOF asymmetrical parallel mechanisms using passive constraining limb: (a) 5-SPS&PPRU and (b) 5-UCU&RRRC. C: cylindrical joint, P: prismatic joint, R: revolute joint, S: spherical joint, U: universal joint.

Fig.10 Construction of the L_1F1C-limb. L_1F1C: passive constraining limb provides both a constraint force and a constraint couple. DOF: degree-of-freedom.

Tab.1 Feasible L_1F1C limbs for side-mounting

Fig.11 Two examples of 2R2T 4-degree-of-freedom asymmetrical parallel mechanisms using the L_1F1C limb: (a) 3-UCU&PPU and (b) 3-SPS&RRRR. C: cylindrical joint, P: prismatic joint, R: revolute joint, S: spherical joint, U: universal joint.

Fig.12 Conceptual diagram of the side-mounted (3-UPRU&RRRR)&R hybrid leg mechanism. P: prismatic joint, P: prismatic joint with actuation, R: revolute joint, R: revolute joint with actuation, U: universal joint.

Fig.13 Schema diagram of the 3-UPRU&RRRR parallel mechanism (left) and the (3-UPRU&RRRR)&R hybrid leg mechanism (right). P: prismatic joint with actuation, R: revolute joint, R: revolute joint with actuation, U: universal joint.

Fig.14 Schematic diagram of the limbs in the 3-UPRU&RRRR parallel mechanism: (a) UPRU limb and (b) RRRR limb.

Fig.15 Singularity configurations of the side-mounted 3-UPRU&RRRR mechanism: (a) three UPRU limbs are parallel, (b) the axial vectors of all limbs intersect at one point, (c) the rank of the of all limbs’ axial vectors is less than four, (d) the 1st and 2nd joints of the RRRR limb are concentric, (e) O_BD′ and D′E′ are stretching collinear, and (f) O_BD′ and D′E′ are folding collinear.

Structural parameter	Value/mm
$? B O B A i$	$? B O B A 1 = [175, 0, 170.5] T$ , $? B O B A 2 = [? 175, 0, 170.5] T$ , $? B O B A 3 = [0, 420, 170.5] T$
$\| ? O B D ′ \|$	220
$\| ? D ′ E ′ \|$	225
$\| ? E ′ O P \|$	582
$? P O P B i$	$? P O P B 1 = [? 803, ? 80, 21.5] T$ , $? P O P B 2 = [803, ? 80, 21.5] T$ , $? P O P B 3 = [0, 160, 21.5] T$

Tab.2 Important structural parameters of the 3-UPRU&RRRR mechanism

Fig.16 Velocity mapping analysis of the side-mounted 3-UPRU&RRRR mechanism: (a) initial configuration of the mechanism, (b) desired trajectory of the moving platform, (c) position and orientation curve of the moving platform, (d) velocity curve of the moving platform, (e) driver position curve, (f) driver actual speed curve, (g) driver calculated speed curve, and (h) error between the actual and calculated driver speed. q_i (i = 1,2,…,6) are the six virtual drive joints considered when calculating the driving speed through the Jacobian matrix, in which q₁ corresponds to l₁, q₂ to l₂, q₃ to l₃, and q₄ to θ, while q₅ and q₆ do not correspond to any drive joints in the mechanism, they are only for analysis.

Fig.17 Workspace of the side-mounted 3-UPRU&RRRR parallel mechanism: (a) α′ = 0, γ′ = 0 in spatial view, (b) α′ = 0, γ′ = 0 in plane, (c) α′ = ?10°, γ′ = 0 in plane, (d) α′ = 20°, γ′ = 0 in plane, (e) α′ = 0, γ′ = ?30° in plane, and (f) α′ = 0, γ′ = 30° in plane.

Fig.18 Workspace of the 3-UPRU&RRRR hybrid leg mechanism: (a) in spatial view, (b) in spatial view without limitations, (c) top view, (d) left view, and (e) front view.

Fig.19 Dexterity map of the 3-UPRU&RRRR mechanism in initial posture and ultimate posture: (a) α′ = ?10°, γ′ = 0°, (b) α′ = 0°, γ′ = 0°, (c) α′ = 20°, γ′ = 0°, (d) α′ = 0°, γ′ = ?30°, and (e) α′ = 0°, γ′ = 30°. LCI: local condition index.

Fig.20 Global conditioning index of the 3-UPRU&RRRR mechanism under different postures: (a) η_ω, (b) η_v, and (c) boxplot of η_ω and η_v.

Fig.21 Static simulation of the side-mounted leg under different common force conditions: (a) condition 1, (b) condition 2, (c) condition 3, and (d) condition 4.

Fig.22 Driving force and torque of actuators of the side-mounted leg in static simulation: (a) condition 1, (b) condition 2, (c) condition 3, and (d) condition 4.

Fig.23 Kinetostatic simulation for a composite translation of the leg: (a) timing diagram, (b) driving force and torque of actuator, and (c) position of actuator.

Fig.24 Kinetostatic simulation for posture adjustment of the leg: (a) timing diagram, (b) driving force and torque of actuator, and (c) position of actuator.

Fig.25 Model of the quadruped rover BJTUBOT using the novel side-mounted 3-UPRU&RRRR&R hybrid leg mechanism.

Fig.26 Different forms of motion of the side-mounted leg mechanism in the virtual model of BJTUBOT for simulation: (a) MF1, (b) MF2, (c) MF3, (d) MF4, and (e) MF5.

Fig.27 Omnidirectional locomotion of the rover: (a) t = 0 s, (b) t = 1 s, (c) t = 3 s, (d) t = 5 s, (e) t = 10 s, (f) t = 11 s, (g) t = 13 s, and (h) t = 14 s.

Fig.28 Ackerman steering locomotion of the rover: (a) t = 0 s, (b) t = 5 s, (c) t = 12 s, and (d) t = 22 s. CoM: center of mass.

Fig.29 Pivot steering locomotion of the rover: (a) t = 0 s, (b) t = 1 s, (c) t = 9 s, (d) t = 14 s, (e) t = 20s, (f) t = 25 s, (g) t = 26 s, and (h) t = 28 s.

Fig.30 Active suspension system function of the rover: (a) t = 0 s, (b) t = 4 s without suspension, (c) t = 8 s without suspension, (d) t = 11 s without suspension, (e) t = 4 s with suspension, (f) t = 8 s with suspension, and (g) t = 11 s with suspension. CoM: center of mass.

Fig.31 Basic walking locomotion of the rover: (a) t = 0 s, (b) t = 1 s, (c) t = 2 s, (d) t = 3 s, (e) t = 4 s, (f) t = 5 s, (g) t = 6 s, (h) t = 7 s, (i) t = 8 s, (j) t = 9 s, (k) t = 10 s, and (l) t = 11 s.

Fig.32 Crawling locomotion of the rover: (a) t = 0 s, (b) t = 4 s, (c) t = 7 s, and (d) t = 10 s. CoM: center of mass.

Fig.33 Jumping locomotion of the rover: (a) t = 0 s, (b) t = 1 s, (c) t = 2 s, (d) t = 4 s, (e) t = 8 s, (f) t = 10 s, (g) t = 12 s, and (h) t = 14 s. CoM: center of mass.

Abbreviations
C	Cylindrical joint
COM	Center of mass
DOF	Degree-of-freedom
GCI	Global conditioning index
L_1F1C	Passive constraining limb provides both a constraint force and a constraint couple
LCI	Local condition index
LT	Locomotion task
MF	Motion form
P	Prismatic joint
P	Prismatic joint with actuation
R	Revolute
R	Revolute joint with actuation
R₁	Type-1 revolute joint
R₂	Type-2 revolute joint
S	Spherical joint
T	Translation
U	Universal joint
Variables
a_i	Vector $? O B A i$
$B a i$	a_i in the base coordinate
$‖ A ‖$	Frobenius norm of matrix A_m×n
b_i	Vector $? O P B i$
$B b i$	b_i in the base coordinate
d	Dimension of the wrench system
d	Vector $? C D$
d'	Vector $? O B D ′$
e	Vector $? D E$
e'	Vector $? O B E ′$
f	Constraint force
F	A support force or static friction
F₁	Support force along the y-axis
F₂, F₃	Static frictions along the x- and z-axis, respectively
F₁(t), F₂(t), F₃(t)	Time-varying functions of F₁, F₂, and F₃, respectively
G	Gravity force
h	Height of the side-mounted base along the y-axis
J	Jacobian matrix of the 3-UPRU&RRRR parallel mechanism
J_{c_RRRR}	Constraint Jacobian for the RRRR limbs
J_{k_UPRU}	Actuation Jacobian for the UPRU limbs
$J v$	Linear velocity mapping part of J
$J ω$	Angular velocity mapping part of J
k	Vector $? O P K$
$B k$	k in the base coordinate
$P k$	k in the moving platform coordinate
$W k$	k in the wheel coordinate
$k (J)$	Condition number of the Jacobian matrix J
l_i	Length of the vector $? A i B i$
$l i$	Vector $? A i B i$
$l ˙ i$	Linear velocity of the P joint
l_max	Maximum length of the P joint
l_min	Minimum length of the P joint
$L ˙$	Vector which contains all velocities of the P joints
m	Constraint couple
M	Torque caused by friction
$M (t)$	Time-varying function of M
N	Support force
O_B	Original point of the base coordinate system
O_P	Original point of the moving platform coordinate system
O_W	Original point of the leg-end coordinate system
p	Vector $? O B O P$
p_W	Vector $? O B O W$
$B p W$	Position vector of the leg-end in the base coordinate
$q ˙ i j$	Intensity of the jth joint in the ith limb
$Q ˙$	Velocity vector of all joints
r	Location vector of the twist $
r_i	Location vector of the ith twist or screw $_i
$r d r$	Location vector of the dth constraint screw $S / d r$
$B R P$	Rotation matrix from the moving platform frame to the base frame
$B R W$	Rotation matrix from the leg-end frame to the base frame
$P R W$	Rotation matrix from the leg-end frame to the moving platform frame
$W R P$	Inverse of $P R W$
s	Direction vector of the twist $
s_i	Direction vector of the ith twist or screw $_i
$s i j$	Direction vector of $S / i j$
$s d r$	Direction vector of the dth constraint screw $S / d r$
t	Time
u_li	Unit vector of P joint
u_d	Unit vector of the linkage CD
$B u l i$	u_liin the base coordinate
$B v$	Vector v in the base frame
$B v O P$	Linear velocity of moving platform
$P v$	Vector v in the moving platform frame
$W v$	Vector v in the leg-end frame
w	Width of the side-mounted base along the x-axis
w	Vector $? K O W$
WS	Workspcae
x	x-axis of the base coordinate system
$x ˙$	Velocity in the x direction
x_P	x-axis of the moving platform coordinate system
$x (t)$	Position function on the x-axis
$X ˙$	Velocity of moving platform
y	y-axis of the base coordinate system
$y ˙$	Velocity in the y direction
y_P	y-axis of the moving platform coordinate system
$y (t)$	Position function on the y-axis
z	z-axis of the base coordinate system
z_P	z-axis of the moving platform coordinate system
α	Attitude angle about the x-axis
$α ˙$	Velocity of α
$α ′$	Rotation angle about the x_P-axis
$α ′ (t)$	Orientation function about α'
β	Attitude angle about the y-axis
$β ˙$	Velocity of β
γ	Attitude angle about the z-axis
$γ ˙$	Velocity of γ
γ'	Rotation angle about the z_P-axis
$γ ′ (t)$	Orientation function about γ'
$η i$	Value of GCI
$η v$ , $η ω$	GCI of the linear and angular motions, respectively
θ	Rotation angle of the R joint in the RRRR limb
$θ ˙$	Angular velocity of active R joint
$θ ˙ i n$	Intensity or angular velocity of the passive joints
φ	Rotation angle of the steering R joint connected with the leg-end
$B ω$	Angular velocity of moving platform
$δ Q ˙$	Driving velocity deviation
$	A screw or a twist
$^r	A wrench system
$_i	ith twist or screw
$S / i j$	Unit screw of the jth joint in the ith limb
$S / P$	Instantaneous twist of the moving platform
$S / d r$	dth constraint screw
$S / i r$	Reciprocal screws for the ith UPRU limb
$δ S / P$	Velocity deviation of the moving platfrom

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