Abbreviations |
BCS | Body coordinate system |
CBC | Centroid balance control |
CI | Convolution interpolation |
CNN | Convolutional neural network |
CoM | Center of mass |
GRF | Ground reaction force |
IMU | Inertial measurement unit |
LCS | Leg coordinate system |
NDCI | Normalized derivable convolution interpolation |
PCI | Pyramid convolution interpolation |
RGBD | Red, green, blue, depth |
SLIP | Spring-loaded inverted pendulum |
WCS | World coordinate system |
Variables |
A | Merged matrix for CBC |
b, f | Merged vectors for CBC |
ci, j | Center of the pixel (i, j) on the map |
C | Matrix of friction cone and upper and lower bound constraints for GRF |
E | Identity matrix |
fi | GRF of the ith leg |
| Virtual feedforward force of the ith leg in the LCS |
flim | Friction cone and upper and lower bound constraints for GRF |
| Virtual force of the supporting legs |
| Virtual force of the ith supporting leg |
, | Virtual feedforward forces of the ith supporting leg and swing leg in the LCS, respectivley |
g | Local gravitational acceleration |
hz | Step height that needs to be raised in the middle of the swing process |
h | Merged vector of hi |
h* | Horizontal component of optimized footholds |
hi | Horizontal component of pi |
| Optimized horizontal coordinate of the foothold |
hpre | Merged vector of hpre,i |
hpre,i | Horizontal component of preplanned foothold |
Δh | Intermediate variable of the difference between the optimized footholds and the preplanned footholds |
Δhlim | Maximum amount of adjustment Δpi allowed |
nΔh | nth element of Δh |
i | Index for the item, such as the ith leg, or the (i, j) pixel |
BI | Inertia matrix of the body in the BCS |
j | Index for the item, such as the (i, j) pixel |
Ji | Jacobian matrix of the ith leg |
J (hi) | Terrain cost at the position of hi |
J (ph) | Cost function for NDCI |
| Partial derivative of the cost function for NDCI |
, | Joint damping and stiffness matrices, respectively |
, | Damping and stiffness of body position, respectively |
, | Damping and stiffness of body attitude, respectively |
, | Damping and stiffness of the leg, respectively |
Bli | Tip position in the BCS |
, | Position and velocity of the ith leg in the LCS, respectively |
, | Target position and velocity of the ith leg in the LCS, respectively |
m, n | Set numbers of pixels that need to be involved |
| Position of the robot body |
, , | Target position, velocity, and acceleration of the robot’s CoM, respectively |
ph | Horizontal coordinate of the position of the selected point on the whole map |
| Foothold point of the ith leg |
, , | x, y, and z components of the foothold , respectively |
| Optimized footholds |
| Height components at the position on the heightmap |
| Preplanning of the target foothold |
| Swing trajectory of the ith leg |
, | Estimated position and velocity of the robot, respectively |
| Correction amount on the original body trajectory |
Δpi | Foothold adjustment for the ith leg |
Bphip,i | Position of the ith leg’s hip joint in the BCS |
, | Current angle and angular velocity read by the ith encoder, respectively |
, | Target angle and angular velocity of the ith joint, respectively |
R | Current attitude matrix |
| Desired attitude of the body |
| Estimated current attitude of the robot |
| Set of real number |
S | Selection matrix for CBC |
tsw,i | Time when the ith leg enters the swing phase |
Tst, Tsw | Duration of the standing and swing phase, respectively |
v | Velocity of the body |
vd | Desired velocity of the body |
| Speed correction |
xh | x component of the horizontal coordinate of the position of the selected point on the heightmap |
xi,j | x component of the center of the pixel (i, j) on the map |
yh | y component of the horizontal coordinate of the position of the selected point on the heightmap |
yi,j | y component of the center of the pixel (i, j) on the map |
W, D, Q | Weighting matrices for footholds optimization |
α, β, γ, | Weighting factors for footholds optimization |
δ | Weighting factor for CBC |
σx, σy | Smoothing factor in x and y directions, respectively |
ϕ, θ, ψ | Euler angle roll, pitch, and yaw measured by the IMU |
μ | Number of legs in contact |
| Control torque of the ith joint |
ξi | Swing phase |
| Angular velocity of the body |
, | Target angular velocity and acceleration of the robot, respectively |
* | Symbol maps from a vector () to an antisymmetric matrix () |