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## Combination of the Direct and the Indirect Method

Following [27] and [28] both methods are combined as follows: The direct collocation method is applied with a poor initial guess of the solution , , i. e., with an initial trajectory that interpolates the given values at initial and final time linearly. The obtained (suboptimal) solution provides reliable estimates of the state and adjoint variables and of the switching structure of state and control constraints (cf. [26], [27]). With this guess the multiple shooting method is applied to the multi-point boundary value problem resulting from the necessary conditions of optimality. For the derivation of the necessary conditions we used the symbolic computation method MAPLE [6]: The equations of motion are explicitly given in the appendix of [19] in the form of Eq. (1). First the explicit inverse of the mass matrix is computed. Then the partial derivatives of each component of the vector function with respect to , , i=1,2,3, are derived. As output of the MAPLE program we obtain a FORTRAN code for , the adjoint differential equations, the Hamiltonian function and the formulae for the boundary controls from or in the state constrained case. The resulting FORTRAN codes for the inverse mass matrix are about 630 lines and for the adjoint differential equations are about 3350 lines long although in each step of the derivation several optimization strategies are applied in MAPLE to simplify the resulting formulae. For more than three degrees of freedom it might be more efficient to use automatic differentiation [10] and to make even more use of the special structure of the robotic dynamics in order to keep the number of the resulting arithmetic operations as small as possible.

Figure 2: Combination of direct, indirect and symbolic methods in robot trajectory optimization.

Next: Numerical Results Up: Numerical Methods Previous: Direct Collocation Method

Oskar von Stryk
Fri Apr 5 21:57:02 MET DST 1996