Abbreviations |
C | Cylindrical joint |
COM | Center of mass |
DOF | Degree-of-freedom |
GCI | Global conditioning index |
L1F1C | 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 |
ai | Vector |
| ai in the base coordinate |
| Frobenius norm of matrix Am×n |
bi | Vector |
| bi in the base coordinate |
d | Dimension of the wrench system |
d | Vector |
d' | Vector |
e | Vector |
e' | Vector |
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 |
Jc_RRRR | Constraint Jacobian for the RRRR limbs |
Jk_UPRU | Actuation Jacobian for the UPRU limbs |
| Linear velocity mapping part of J |
| Angular velocity mapping part of J |
k | Vector |
| k in the base coordinate |
| k in the moving platform coordinate |
| k in the wheel coordinate |
| Condition number of the Jacobian matrix J |
li | Length of the vector |
| Vector |
| Linear velocity of the P joint |
lmax | Maximum length of the P joint |
lmin | Minimum length of the P joint |
| Vector which contains all velocities of the P joints |
m | Constraint couple |
M | Torque caused by friction |
| Time-varying function of M |
N | Support force |
OB | Original point of the base coordinate system |
OP | Original point of the moving platform coordinate system |
OW | Original point of the leg-end coordinate system |
p | Vector |
pW | Vector |
| Position vector of the leg-end in the base coordinate |
| Intensity of the jth joint in the ith limb |
| Velocity vector of all joints |
r | Location vector of the twist $ |
ri | Location vector of the ith twist or screw $i |
| Location vector of the dth constraint screw |
| Rotation matrix from the moving platform frame to the base frame |
| Rotation matrix from the leg-end frame to the base frame |
| Rotation matrix from the leg-end frame to the moving platform frame |
| Inverse of |
s | Direction vector of the twist $ |
si | Direction vector of the ith twist or screw $i |
| Direction vector of |
| Direction vector of the dth constraint screw |
t | Time |
uli | Unit vector of P joint |
ud | Unit vector of the linkage CD |
| uli in the base coordinate |
| Vector v in the base frame |
| Linear velocity of moving platform |
| Vector v in the moving platform frame |
| Vector v in the leg-end frame |
w | Width of the side-mounted base along the x-axis |
w | Vector |
WS | Workspcae |
x | x-axis of the base coordinate system |
| Velocity in the x direction |
xP | x-axis of the moving platform coordinate system |
| Position function on the x-axis |
| Velocity of moving platform |
y | y-axis of the base coordinate system |
| Velocity in the y direction |
yP | y-axis of the moving platform coordinate system |
| Position function on the y-axis |
z | z-axis of the base coordinate system |
zP | z-axis of the moving platform coordinate system |
α | Attitude angle about the x-axis |
| Velocity of α |
| Rotation angle about the xP-axis |
| Orientation function about α' |
β | Attitude angle about the y-axis |
| Velocity of β |
γ | Attitude angle about the z-axis |
| Velocity of γ |
γ' | Rotation angle about the zP-axis |
| Orientation function about γ' |
| Value of GCI |
, | GCI of the linear and angular motions, respectively |
θ | Rotation angle of the R joint in the RRRR limb |
| Angular velocity of active R joint |
| Intensity or angular velocity of the passive joints |
φ | Rotation angle of the steering R joint connected with the leg-end |
| Angular velocity of moving platform |
| Driving velocity deviation |
$ | A screw or a twist |
$r | A wrench system |
$i | ith twist or screw |
| Unit screw of the jth joint in the ith limb |
| Instantaneous twist of the moving platform |
| dth constraint screw |
| Reciprocal screws for the ith UPRU limb |
| Velocity deviation of the moving platfrom |