lc | 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 |
lt | 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 |
ls | Length of shin | \(\sum_{{\text{B}}}\) | Body coordinate system of the robot |
mc | Mass of coxa | \(\sum_{{\text{G}}}\) | Ground coordinate system |
mt | Mass of thigh | \({\varvec{F}}_{{\text{B}}}\) | Principal vectors, including \(F_{{{\text{B}}x}}\), \(F_{{{\text{B}}y}}\), and \(F_{{{\text{B}}z}}\) |
ms | Mass of shin | \({\varvec{M}}_{{\text{B}}}\) | Principal moment, including \(M_{{{\text{B}}x}}\), \(M_{{{\text{B}}y}}\), and \(M_{{{\text{B}}z}}\) |
Ic | Moment of inertia for the coxa | \(\sum_{{f_{i} }}\) | Foot coordinate system of the leg i, it has the same positive direction with the \(\sum_{{A_{i} }}\) |
It | Moment of inertia for the thigh | \(\sum_{{o_{i} }}\) | Foot coordinate system of the leg i, it has the same positive direction with the \(\sum_{{\text{B}}}\) |
Is | 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 sk of the support phase in the \(\sum_{{o_{k} }}\) |
βR | Duty ratio | \(F_{x}^{{\left( {f_{k} } \right)}}\), \(F_{y}^{{\left( {f_{k} } \right)}}\), \(F_{z}^{{\left( {f_{k} } \right)}}\) | Foot forces of leg sk of the support phase in the \(\sum_{{f_{k} }}\) |
Ai | Abductor joint of the leg i | \({}^{{\text{B}}}{\varvec{P}}_{{\text{F}}}^{{\left( {s_{k} } \right)}}\) | Position vector of foot end of the leg sk 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 \({}^{{\text{B}}}P_{{{\text{F}}z}}^{{\left( {s_{k} } \right)}}\) |
Hi | Hip joint of the leg i | \(\sum_{{{0}_{i} }}\) | Coordinate system connecting the leg i and the body |
Ki | Knee joint of the leg i | Δθi | Initial angle of abductor joint between 0º line and leg i |
βi′ | Angle between the coxa and the thigh of leg i | θi | Rotation angle of abductor joint based on the initial angle Δθi of leg i |
βi | Angle between the coxa and the shin of leg i | Φi | Rotation angle between the \(\sum_{{{0}_{i} }}\) and the \(\sum_{{\text{B}}}\) for the leg i |
γi | 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,\(_{0}^{{\text{B}}} {\varvec{R}}^{\left( i \right)} \in {\varvec{R}}^{3 \times 3}\) |
su | 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,\({}_{{\text{A}}}^{{\text{B}}} {\varvec{R}}^{\left( i \right)} \in {\varvec{R}}^{3 \times 3}\) |
tr | 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 te |
\(\user2{\hat{\varvec{M}}}_{i}\) | 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 sk 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 sk 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 sk in view of the foot force | Lpi | 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 sk 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 sk in view of the movement of robot | hi | 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 sk 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 sk | \({}^{{\text{G}}}{\varvec{P}}_{{\text{F}}}^{\left( i \right)}\) | 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 sk | \({}^{{\text{G}}}{\varvec{P}}_{0}^{\left( i \right)}\) | 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 sk | \({}^{0}{\varvec{P}}_{{\text{F}}}^{\left( i \right)}\) | 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 te | \({}^{{\text{B}}}{\varvec{P}}_{0}^{\left( i \right)}\) | 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 te | hTB | Initial lifting height of leg in the transfer phase |
hT-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 \(\sum_{{\text{B}}}\) |
rbp | Effective radius of the bearing platform | μ | Coefficient of ground static friction |
su, sd | 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 |