Table 4 Nomenclature of variables

l_c

Length of coxa

\(x_{2}^{\left( i \right)}\), \(y_{2}^{\left( i \right)}\),\(z_{2}^{\left( i \right)}\)

Components of hip joint coordinate system of the leg i

l_t

Length of thigh

\(x_{3}^{\left( i \right)}\), \(y_{3}^{\left( i \right)}\),\(z_{3}^{\left( i \right)}\)

Components of knee joint coordinate system of the leg i

l_s

Length of shin

\(\sum_{{\text{B}}}\)

Body coordinate system of the robot

m_c

Mass of coxa

\(\sum_{{\text{G}}}\)

Ground coordinate system

m_t

Mass of thigh

\({\varvec{F}}_{{\text{B}}}\)

Principal vectors, including \(F_{{{\text{B}}x}}\), \(F_{{{\text{B}}y}}\), and \(F_{{{\text{B}}z}}\)

m_s

Mass of shin

\({\varvec{M}}_{{\text{B}}}\)

Principal moment, including \(M_{{{\text{B}}x}}\), \(M_{{{\text{B}}y}}\), and \(M_{{{\text{B}}z}}\)

I_c

Moment of inertia for the coxa

\(\sum_{{f_{i} }}\)

Foot coordinate system of the leg i, it has the same positive direction with the \(\sum_{{A_{i} }}\)

I_t

Moment of inertia for the thigh

\(\sum_{{o_{i} }}\)

Foot coordinate system of the leg i, it has the same positive direction with the \(\sum_{{\text{B}}}\)

I_s

Moment of inertia for the shin

\({}^{{\text{B}}}F_{x}^{{\left( {s_{k} } \right)}}\), \({}^{{\text{B}}}F_{y}^{{\left( {s_{k} } \right)}}\),\({}^{{\text{B}}}F_{z}^{{\left( {s_{k} } \right)}}\)

Foot forces of leg s_k of the support phase in the \(\sum_{{o_{k} }}\)

β_R

Duty ratio

\(F_{x}^{{\left( {f_{k} } \right)}}\), \(F_{y}^{{\left( {f_{k} } \right)}}\), \(F_{z}^{{\left( {f_{k} } \right)}}\)

Foot forces of leg s_k of the support phase in the \(\sum_{{f_{k} }}\)

A_i

Abductor joint of the leg i

\({}^{{\text{B}}}{\varvec{P}}_{{\text{F}}}^{{\left( {s_{k} } \right)}}\)

Position vector of foot end of the leg s_k in the \(\sum_{{\text{B}}}\), including \({}^{{\text{B}}}P_{{{\text{F}}x}}^{{\left( {s_{k} } \right)}}\), \({}^{{\text{B}}}P_{{{\text{F}}y}}^{{\left( {s_{k} } \right)}}\), and \({}^{{\text{B}}}P_{{{\text{F}}z}}^{{\left( {s_{k} } \right)}}\)

H_i

Hip joint of the leg i

\(\sum_{{{0}_{i} }}\)

Coordinate system connecting the leg i and the body

K_i

Knee joint of the leg i

Δθ_i

Initial angle of abductor joint between 0º line and leg i

β_i′

Angle between the coxa and the thigh of leg i

θ_i

Rotation angle of abductor joint based on the initial angle Δθ_i of leg i

β_i

Angle between the coxa and the shin of leg i

Φ_i

Rotation angle between the \(\sum_{{{0}_{i} }}\) and the \(\sum_{{\text{B}}}\) for the leg i

γ_i

Angle between the thigh and the shin of leg i

\(_{0}^{{\text{B}}} {\varvec{R}}^{\left( i \right)}\)

Rotation matrix from the \(\sum_{{{0}_{i} }}\) to the \(\sum_{{\text{B}}}\) for the leg i,\(_{0}^{{\text{B}}} {\varvec{R}}^{\left( i \right)} \in {\varvec{R}}^{3 \times 3}\)

s_u

Leg u in the support phase

\({}_{{\text{A}}}^{{\text{B}}} {\varvec{R}}^{\left( i \right)}\)

Rotation matrix from the \(\sum_{{A_{i} }}\) to the \(\sum_{{\text{B}}}\) for the leg i,\({}_{{\text{A}}}^{{\text{B}}} {\varvec{R}}^{\left( i \right)} \in {\varvec{R}}^{3 \times 3}\)

t_r

Leg r in the transfer phase

\({}_{{\text{o}}}^{{\text{f}}} {\varvec{R}}^{\left( i \right)}\)

Rotation matrix from the \(\sum_{{o_{i} }}\) to the \(\sum_{{f_{i} }}\),\({}_{{\text{o}}}^{{\text{f}}} {\varvec{R}}^{\left( i \right)} \in {\varvec{R}}^{3 \times 3}\)

\(\sum_{{A_{i} }}\)

Abductor joint coordinate system of the leg i, including \(x_{1}^{\left( i \right)}\), \(y_{1}^{\left( i \right)}\), and \(z_{1}^{\left( i \right)}\)

\({}_{{\text{A}}}^{{0}} {\varvec{R}}^{\left( i \right)}\)

Rotation matrix from the \(\sum_{{A_{i} }}\) to the \(\sum_{{{0}_{i} }}\),\({}_{{\text{A}}}^{{0}} {\varvec{R}}^{\left( i \right)} \in {\varvec{R}}^{3 \times 3}\)

\(\user2{\tilde{\varvec{M}}}_{i}\)

Vector of articulated torque of the leg i, \(\space \user2{\tilde{\varvec{M}}}_{i} \in {\varvec{R}}^{3 \times 1}\)

\(M_{{\text{K}}}^{{\left( {t_{e} } \right)}}\)

Torque of knee joint of the swinging leg t_e

\(\user2{\hat{\varvec{M}}}_{i}\)

Vector of articulated torque from the dynamics item of the leg i, \(\space \user2{\hat{\varvec{M}}}_{i} \in {\varvec{R}}^{3 \times 1}\)

\(M_{{\text{A}}}^{\left( i \right)}\)

Final output torque of servo motor for the abductor joint of leg i

\(\user2{\hat{\varvec{M}}}_{{{\text{F}}i}}\)

Vector of articulated torque from the foot force of leg i, \(\space \user2{\hat{\varvec{M}}}_{{{\text{F}}i}} \in {\varvec{R}}^{3 \times 1}\)

\(M_{{\text{H}}}^{\left( i \right)}\)

Final output torque of servo motor for the hip joint of leg i

\(\left( {{\varvec{J}}_{i} \left( {\varvec{q}} \right)} \right)^{{\text{T}}}\)

Transpose of velocity Jacobian matrix, \(\space \left( {{\varvec{J}}_{i} \left( {\varvec{q}} \right)} \right)^{{\text{T}}} \in {\varvec{R}}^{3 \times 3}\)

\(M_{{\text{K}}}^{\left( i \right)}\)

Final output torque of servo motor for the knee joint of leg i

\({}^{{\text{B}}}{\varvec{F}}_{i}\)

Foot force vector of the leg i in the coordinate system \(\sum_{{\text{B}}}\) or \(\sum_{{o_{i} }}\), \(\space {}^{{\text{B}}}{\varvec{F}}_{i} \in {\varvec{R}}^{3 \times 1}\)

\(K_{{{\text{MA}}}}^{\left( i \right)}\), \(K_{{{\text{MH}}}}^{\left( i \right)}\), \(K_{{{\text{MK}}}}^{\left( i \right)}\)

Torque constants of servo motors for the abductor joint, hip joint, and knee joint of leg i

\(E^{\left( i \right)}\)

Kinetic energy of the leg i

\(i_{{{\text{CA}}}}^{\left( i \right)}\), \(i_{{{\text{CH}}}}^{\left( i \right)}\), \(i_{{{\text{CK}}}}^{\left( i \right)}\)

Currents of servo motors for the abductor joint, hip joint, and knee joint of leg i

\(P^{\left( i \right)}\)

Potential energy of the leg i

\(i_{{\text{A}}}^{\left( i \right)}\), \(i_{{\text{H}}}^{\left( i \right)}\), \(i_{{\text{K}}}^{\left( i \right)}\)

Transmission ratios for the abductor joint, hip joint, and knee joint of leg i

\(L^{\left( i \right)}\)

Lagrange function of the leg i

\(\eta_{{\text{A}}}^{\left( i \right)}\), \(\eta_{{\text{H}}}^{\left( i \right)}\), \(\eta_{{\text{K}}}^{\left( i \right)}\)

Transmission efficiencies for the abductor joint, hip joint, and knee joint of leg i

\({\varvec{D}}_{i} \left( {\varvec{q}} \right)\)

Inertial matrix of the joint of leg i, \(\space {\varvec{D}}_{i} \left( {\varvec{q}} \right) \in {\varvec{R}}^{3 \times 3}\)

\(P_{{\text{A}}}^{\left( i \right)}\), \(P_{{\text{H}}}^{\left( i \right)}\), \(P_{{\text{K}}}^{\left( i \right)}\)

Power consumption of the servo motors for the abductor joint, hip joint, and knee joint of leg i

\({\varvec{h}}_{i} \left( {{\varvec{q}},\dot{\user2{q}}} \right)\)

Force vector of the leg i including the centrifugal inertial force and Coriolis inertial force, \(\space {\varvec{h}}_{i} \left( {{\varvec{q}},\dot{\user2{q}}} \right) \in {\varvec{R}}^{3 \times 3}\)

\( U_{{\text{A}}}^{\left( i \right)}\), \(U_{{\text{H}}}^{\left( i \right)}\), \(U_{{\text{K}}}^{\left( i \right)}\)

Voltages of servo motors for the abductor joint, hip joint, and knee joint of leg i

\({\varvec{g}}_{i} \left( {\varvec{q}} \right)\)

Gravity vector of the leg i, \(\space {\varvec{g}}_{i} \left( {\varvec{q}} \right) \in {\varvec{R}}^{3 \times 1}\)

\(P_{{{\text{sum}}}}^{\left( i \right)}\)

Power consumption of the leg i

\(M_{{{\text{SA}}}}^{{\left( {s_{k} } \right)}}\)

Torque of abductor joint of the support leg s_k in view of the foot force

\(P_{{{\text{sum}}}}\)

Power consumption of robot mobile system

\(M_{{{\text{SH}}}}^{{\left( {s_{k} } \right)}}\)

Torque of hip joint of the support leg s_k in view of the foot force

s

Step pitch of robot

\(M_{{{\text{SK}}}}^{{\left( {s_{k} } \right)}}\)

Torque of knee joint of the support leg s_k in view of the foot force

L_pi

Span, projection length of the leg i in the coordinate system of abductor joint

\(M_{{{\text{DA}}}}^{{\left( {s_{k} } \right)}}\)

Torque of abductor joint of the support leg s_k in view of the movement of robot

L′_pi

Initial span of the leg i when the robot is about to walk.

\(M_{{{\text{DH}}}}^{{\left( {s_{k} } \right)}}\)

Torque of hip joint of the support leg s_k in view of the movement of robot

h_i

Distance from the coordinate origin of \(\sum_{{A_{i} }}\) to the ground

\(M_{{{\text{DK}}}}^{{\left( {s_{k} } \right)}}\)

Torque of knee joint of the support leg s_k in view of the movement of robot

h

Body height of robot

\(M_{{\text{A}}}^{{\left( {s_{k} } \right)}}\)

Total torque of abductor joint of the support leg s_k

\({}^{{\text{G}}}{\varvec{P}}_{{\text{F}}}^{\left( i \right)}\)

Position vector from the foot of leg i to the ground coordinate system \(\sum_{{\text{G}}}\)

\(M_{{\text{H}}}^{{\left( {s_{k} } \right)}}\)

Total torque of hip joint of the support leg s_k

\({}^{{\text{G}}}{\varvec{P}}_{0}^{\left( i \right)}\)

Position vector from the coordinate system \(\sum_{{{0}_{i} }}\) to the ground coordinate system \(\sum_{{\text{G}}}\)

\(M_{{\text{K}}}^{{\left( {s_{k} } \right)}}\)

Total torque of knee joint of the support leg s_k

\({}^{0}{\varvec{P}}_{{\text{F}}}^{\left( i \right)}\)

Foot position vector of the leg i in the coordinate system \(\sum_{{{0}_{i} }}\)

\(M_{{\text{A}}}^{{\left( {t_{e} } \right)}}\)

Torque of abductor joint of the swinging leg t_e

\({}^{{\text{B}}}{\varvec{P}}_{0}^{\left( i \right)}\)

Position vector form the coordinate system \(\sum_{{{0}_{i} }}\) to the body coordinate system \(\sum_{{\text{B}}}\) for the leg i

\(M_{{\text{H}}}^{{\left( {t_{e} } \right)}}\)

Torque of hip joint of the swinging leg t_e

h_TB

Initial lifting height of leg in the transfer phase

h_T-max

Maximum swinging height of leg in the transfer phase

\({\varvec{J}}_{{\text{M}}}\)

Position matrix of foot ends of the support phase in the \(\sum_{{\text{B}}}\)

r_bp

Effective radius of the bearing platform

μ

Coefficient of ground static friction

s_u, s_d

Upper point of foot and nether point of foot when the robot walks

\({}^{{\text{B}}}{\varvec{F}}_{{\text{s}}}\)

Force matrix of the foot in the coordinate system \(\sum_{{\text{B}}}\) for the leg in the support phase