Quadruped Robot Modeling and Numerical Generation of the OpenLoop Trajectory

1
Quadruped Robot Modeling and Numerical Generation
of the Open-Loop Trajectory
Dan Jakeway, Prof. Jongeun Choi, Prof. Ranjan
Mukherjee Michigan State University Electrical
Engineering, Intern Mechanical Engineering,
Advisors

• Stance Leg Coordinates
• The generalized vector of coordinates is q
• The Cartesian coordinates are written in terms
  of the more desirable generalized coordinates,
  Fig. 2 illustrates the positions of the
  generalized coordinates.
• We introduce the auxiliary variable
  to avoid tedious rewriting in
  hand calculations.
• Given the Cartesian coordinates, compute the
  corresponding Jacobian matrices w.r.t generalized
  coordinates

• Introduction
• We model a symmetric quadruped gait for a planar
  robot with actuated hips and knees. The degrees
  of freedom are reduced to two in the stance leg
  to simplify analysis and design of the periodic
  control input of this highly nonlinear state
  space model. We illustrate the methodology in
  software for converting the Cartesian coordinates
  and model parameters of the links centers of
  masses into generalized coordinates to derive the
  mass-inertia matrix, the Coriolis and centripetal
  force matrix, and gravity vector in the standard
  dynamical equations of n-link chains. The
  generation of the numerically integrated open
  loop optimal trajectory, which is identical to
  the feedback linearization of this nonlinear
  system given a reference signal, is outlined for
  the quadruped.

• Optimal Control
• The calculus of variations with the
  Euler-Lagrange equations on the performance index
  integrand is the classical method, but this is
  numerically unstable.
• The Pontryagin Maximum Principle is employed,
  which is a necessary condition for optimality
  (i.e. the optimal trajectory must satisfy the
  Maximum Principle but this isnt sufficient for
  identifying the global optimum)
• We first need to define our performance index
  which is to be minimized with respect to a
  trajectory variation, and we must choose our
  desired time interval.
• The state equations are derived from the
  following
• The co-state differential equations are
• Which are then numerically solved in MATLAB via
  any of a variety of BVP solvers.
• Future Worka singular analytic Jacobian matrix
  may arise for common cofigurations, which
  produces a singularity through the trajectory.
  Complex code and further research can rectify
  this problem.

• Gait
• Fig. 1 is the dorsal (top-down) view of the
  robot
• The legs are numbered 1-4
• The notation C1234 means legs one and two are
  aloft in the swing phase, three and four are in
  stance
• Using this notation, our desired gait is
• C1234? C1234? C1234? C1234? C1234
• The robot will not tip over in the lateral
  direction when lifting one pair of legs because
  we restrict to the planar case, which is
  typically done in experiments anyway (Raibert,
  Legged Robots That Balance, 1986)
• The stance legs are synchronized with each
  other so that on level ground the body doesnt
  dip. This allows us to consider the coordinates
  of only one stance leg in the analysis, because
  the contribution of kinetic energy and velocities
  in the paired leg are identical.

Fig. 1 Dorsal View of Quadruped Model
Fig. 2 Illustration of the State
Space Generalized Coordinates