Posture Prediction and Optimization - Human Motion Simulation: Predictive Dynamics - page 60

3.10 Motion prediction formulation
The optimum motion along a path, given by a set of joint trajectories, is deter-
mined by solving the following optimization problem:

Find: q ð u Þ 5 f q i ð u Þj 1 # i # nDOF g

; :

to minimize: Discomfort etc

(3.22) 2

subject to: jj x

end 2 eff

ð q ð t j ÞÞ x

path

ð t j Þjj

# εð distance to path Þ

P ð i k # q Upper

i

q Lower
i

and ð joint limits Þ

#

where q(u) represents motion profiles for each joint. For this problem in Equation
(3.22) , the feasible space is defined as the set of all solutions q(u) for which every
constraint is satisfied.

3.10.1 Design variables
As previously discussed, the design variables are the (n 1) (nDOF) coefficients,

1

f P ð i Þ
0

P ð i Þ
1

P ð i Þ
n

nDOF. These coefficients shape the B-spline curve
into optimal joint trajectories (which we call joint profiles). The resulting curve
plots joint values (radians) versus time (seconds). The optimization requires an
“initial guess” for the design variables. In this case, the coefficients for the ith
joint are evenly spaced between the initial joint value q init
i

; ; ...; g for 1 i

# #

and the final joint value

q final
i

. Hence, the initial guess for q i (u) becomes:

k
n ð q final

P ð i Þ
k

q init
i

q init
i

5 1 2 Þ (3.23)

i

where there are n 1 coefficients and 0 k n. This guess satisfies the joint

1 # #

limits as long as the initial and final joint values also satisfy joint limits.

However, it does not necessarily satisfy the distance constraint.

3.10.2 Constraints
The first constraint in Equation (3.22) is the distance constraint, which requires
that the end-effector remain in contact with the given Cartesian path. The remain-
ing constraints ascertain that each curve q i (u) lies between the upper and lower lim-
its for that joint. Using the property of B-spline curves, restricting every coefficient
P ð i k to be within the joint limits will guarantee that the entire curve q i (u) lies within
the joint limits. Enforcing these constraints helps ensure that the algorithm does not
result in an unrealistic motion, and as a result will stay on the given path.