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
Introduction
• The generalized vector of coordinates is q
•
T
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.
q  ( , )
  femur angle w.r.t body
  shin angle w.r.t. extended femur
• 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
J 
Gait
• Fig. 1 is the dorsal (top-down) view of the robot
• The legs are numbered 1-4
1
2
3
•
C1234
C1234
C1234
4
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 doesn’t 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.
q
r
Fig. 1
Dorsal View of
Quadruped Model


J 
,
q
r
J 
,
q
• 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
J 
r
,
J 
(r )
q
(r )
• 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
isn’t 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:


D q C q g  u

 x  f ( x, u )
• The co-state differential equations are:
( Jacobian of femur angle)
(Jacobian of shin angle)
q
(r )

J 
(Jacobian of torso angle)
q
D  m[2 J  (q ) J  (q)  2 J (q) J (q)  J (q ) J (q )] 
T
Optimal Control
T
T
 I [2( J  )T J   2( J )T J  ( J )T J ] , mass - inertia matrix
1  d kj d ki d ij 
cijk  


 , d ij  i - j coord. of D above
2  qi q j qk 
1 T
T
H  u u   f ( x, u )
2

H
H
 f 
T
T
T f
 f ( x, u )  x ,

(u      )

x
x
 u 
• Which are then numerically solved in MATLAB via any
of a variety of BVP solvers.
• Future Work—a 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.
Ckj   cijk , the k - j coordinate of the Coriolis matrix
i
T
 P 
P  m[ g (2r  2r  r )] , g (q )    (gravity v ector)
 q 
T
Fig. 2
Illustration of the State Space
Generalized Coordinates