Biped Kinematics and Dynamics

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Biped Kinematics and Dynamics
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Two Link Model
?2
L2
l1 L1/2
?1
L1
l2 L2/2
m1 is mass of Link 1
m2 is mass of Link 2
G1 is centre of mass of Link 1
G2 is centre of mass of Link 2
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Kinematic Model

• Simple model
• d?1/dt, d?2/dt ?1, ?2
• where ?1and ?2 are the angular velocities
  applied to the joints

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Newtonian Dynamic Model

• XG1 l1 sin ?1 XG2 L1 sin ?1 l2 sin ?2
• YG1 l1 cos ?1 YG2 L1 cos ?1 l2 cos ?2
• Joint 2
• m2d2(XG2)/dt2l2 cos ?2 - m2d2(YG2)/dt2l2 sin ?2
  m2g l2 sin ?2 ?2
• Joint 1
• m1d2(XG1)/dt2l1 cos ?1 - m1d2(YG1)/dt2l1 sin ?1
  m2d2(XG2)/dt2L1 cos ?1 - m2d2(YG2)/dt2L1 sin ?1
  m1g l1 sin ?1 m2g L1 sin ?1 ?1

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Joint 2

• ?1(m2 L1 l2cos ?1cos?2 m2 L1 l2sin?1sin ?2)
  ?2(m2 l22 cos2 ?2 m2 l22 sin2 ?2) ?12
  (-m2 L1 l2sin?1cos?2m2L1 l2 cos?1sin?2) ?22
  (-m2 l22sin?2cos?2m2 l22 sin?2cos ?2) m2g l2
  sin ?2 ?2
• so
• ?1(m2 L1 l2 cos (?2 -?1)) ?2(m2 l22) ?12
  (m2 L1 l2 sin(?2 -?1)) ?22 (0)
  m2g l2 sin ?2 ?2

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Joint 1

• ?1(m1 l12 cos2 ?1 m2 L12 cos2 ?1 m1 l12 sin2
  ?1 m2 L12 sin2 ?1)
  ?2(m2 L1 l2 cos ?1 cos
  ?2 m2 L1 l2 sin ?1 sin ?2 ) ?12 (-m2
  l22sin?1cos?2-m2 L22sin?1cos?2m2
  l22sin?1cos?2m2 L22sin?1cos?2)
  ?22 (-m2 L1l2 sin?2cos?1m2 L1l2
  sin?1cos ?2) g sin ?1(m1 l1 m2 L1 )
  ?1
• so
• ?1(m1 l12 m2 L12) ?2(m2 L1 l2cos(?2-?1))
  ?12 (0) ?22 (-m2 L1 l2 sin(?2 -?1))
  g sin ?1(m1 l1 m2 L1 ) ?1

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Full Dynamic Equation

• M(?)? F(?,?)? G(?) T(?)
• where M is the inertia matrix, F the
  coriolis/centripedal matrix, G the gravity vector
  and T the torque vector
• The equations can also be derived using
  Lagrangian techniques.

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General Biped Models

• 4 link planar biped
• can be derived using similar techniques to above
• 7 link 3D biped
• analysis more complex, but can be similarly
  derived