O. von Stryk, R. Bulirsch. Direct and indirect methods for trajectory optimization. Annals of Operations Research 37:357--373, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....original time interval is divided into subintervals in a multiple shooting type approach that provides a source of parallelism. For other approaches, see, e.g. Dickmanns and Well [11] Kraft [20] Hargraves and Paris [19] Pesch [28] Lamour [21] Betts and Huffman [3] von Stryk and Bulirsch [35], Bulirsch et al. 9] von Stryk [34] Betts [2] Brenan [6] Schulz, Bock and Steinbach [30] Tanartkit and Biegler [32] Pantelides, Sargent and Vassiliadis [27] and Gritsis, Pantelides and Sargent [18] This research was partially supported by National Science Foundation grants ....

O. von Stryk and R. Bulirsch, Direct and indirect methods for trajectory optimization, Ann. Oper. Res., 37 (1992), pp. 357--373. 22 L. PETZOLD, J. B. ROSEN, P. E. GILL, L. O. JAY AND K. PARK

....original time interval is divided into subintervals in a multiple shooting type approach that provides a source of parallelism. For other approaches, see, e.g. Dickmanns and Well [11] Kraft [20] Hargraves and Paris [19] Pesch [28] Lamour [21] Betts and Huffman [3] von Stryk and Bulirsch [35], Bulirsch et al. 9] von Stryk [34] Betts [2] Brenan [6] Schulz, Bock and Steinbach [30] Tanartkit and Biegler [32] Pantelides, Sargent and Vassiliadis [27] and Gritsis, Pantelides and Sargent [18] The associated finite dimensional optimization problem is characterized by: a) many ....

O. von Stryk and R. Bulirsch, Direct and indirect methods for trajectory optimization, Ann. Oper. Res., 37 (1992), pp. 357--373. 26 L. PETZOLD, J. B. ROSEN, P. E. GILL, L. O. JAY AND K. PARK

....u(t) t) 0 = r(x(0) x(t f ) t f ) 8) where 0 B x 1 . xn 1 C A = 0 B xn 1 . x 2n 1 C A ; 0 B xn 1 . x 2n 1 C A = M Gamma1 (x) Delta (u h(x; t) 9) Numerical Methods for solving problem (8) can be classified into direct and indirect methods [22]. Indirect methods, such as the multiple shooting method, gradient and min H methods, are mainly based on the Maximum Principle. Here the adjoint variables (t) the adjoint equations = Gamma T f= x etc. are used explicitly. Indirect methods are often cumbersome to apply as they require to ....

.... can be solved efficiently by Sequential Quadratic Programming (SQP) methods [9,19] Most common are direct shooting [1,2,15] and direct collocation methods [10,21] A combination of direct collocation and indirect multiple shooting methods which combines the merits of both has been developed in [21,22,23]. It has been demonstrated that carefully implemented direct methods do provide reliable, easy to use and robust tools for solving optimal control problems if accuracy requirements are not extremely high. For direct transcription methods typically the resulting NLPs are large and sparse and grid ....

[Article contains additional citation context not shown here]

Stryk, O. von; Bulirsch, R.: Direct and indirect methods for trajectory optimization; Annals of Operations Research 37 (1992) 357-373.

....u (t) 0 t t f , must satisfy the Maximum Principle. For optimal off line programming, robust and efficient methods are needed for computing approximations of q and u . A discussion of direct and indirect transcription methods for general trajectory optimization methods can be found in [2,19,21]. Direct transcription (DT) methods are based on a parametrization of control and state variables, e.g. by piecewise polynomial approximations u(t) P N i=1 ff i u (i) t) ff i 2 IR 3 , and q(t) P M j=1 fi j q (j) t) fi j 2 IR 3 , with suitable basis functions u (i) q (j) ....

.... suitable basis functions u (i) q (j) defined on a time grid [2,19] The resulting large, finite dimensional, nonlinearly constrained optimization problems can be solved efficiently by Sequential Quadratic Programming methods [2,5] Most common are direct shooting and direct collocation methods [21]. DT methods are usually more robust and much easier to use than indirect methods which are based on adjoint differential equations [19,21] Several methods have also been developed which are tailored to the robot trajectory optimization problem but are not applicable to general optimal control ....

[Article contains additional citation context not shown here]

Stryk, O. von; Bulirsch, R. (1992) Direct and indirect methods for trajectory optimization, Annals of Operations Research, 37, 357--373.

....to be superior when highly accurate optimal solutions are to be computed and a scrutiny of the necessary conditions of optimal control theory is desired. If these strong requirements are withdrawn for an easier handling of the method, the direct collocation method (see, e.g. 3] 4] 30] 53] [54]) turns out to be more favorable. Inbetween these two methods, the direct simple shooting method [36] 37] and the direct multiple shooting method [5] are established. 2 Hans Josef Pesch The present investigations and the results obtained for various real life optimal control problems (cf. ....

....about the switching structure, i.e. the sequence of the junction points and the associated control laws between these junction points according to necessary conditions. However, knowledge of the switching structure helps to get more accurate results; see [53] In the so called hybrid approach of [54], which amalgamates direct collocation and multiple shooting by estimating the adjoint variables for the indirect approach, the transition from the solution obtained by a direct collocation method to the Solving Optimal Control and Pursuit Evasion Game Problems 3 solution of the multipoint ....

[Article contains additional citation context not shown here]

von Stryk, O. and Bulirsch, R.: Direct and Indirect Methods for Trajectory Optimization, Annals of Operations Research 37 (1992) 357-373.

....tending toward a combination of direct and indirect methods. The result of a direct method together with estimates for the adjoint variables is then used to improve it by an accurate indirect method. This combination and the transition from direct to indirect methods is discussed in [25] [44]. The present paper contributes to the interplay of finite dimensional approximations to (1.1) 1.5) and their solution by nonlinear programming methods. An important question in this interplay consists in the appropriate and efficient calculation of first order derivatives of the objective ....

O. von Stryk and R. Bulirsch, Direct and indirect methods for trajectory optimization, Annals of Operations Research 37 (1992), 357-373.

.... S c (t) L c (t) Y c (t) and I(t) Therefore the goal in the optimal control problem can be formulated as X (t f ) Xm (t f ) 1 Gamma ) Delta p Delta S(t f ) d(t f ) Gamma max : The optimal control problem has been solved with the help of the new collocation method DIRCOL, see von Stryk 1992. The major advantages of this direct method are: ffl Non differentiable model functions, e.g. ae k and ae m , can be handled, since no explicit numerical integration of the differential equations is done. ffl The large domain of convergence enables the computation of the optimal solution even ....

Stryk, von, O. (1992): Direct and Indirect Methods for Trajectory Optimization. Annals of Operations Research 37, pp. 357 -- 373.

....h(x(t) p; u(t) 0; t 0 t t f ; h 2 IR nh : 3) Equations (1) 3) define an optimal control problem. Its solution must satisfy the Maximum Principle [20] As the adjoint (costate) differential equations cannot be computed without enormous efforts in our case, a direct transcription method [2,28] is investigated for computing an approximation of the optimal (open loop) control u : t 0 ; t f ] IR nu and p 2 IR np numerically. By a parameterization of the control variable u( p) p 2 IR n p , the optimal control problem becomes a nonlinear programming problem (NLP) for ....

Stryk, O. von; Bulirsch, R.: Direct and indirect methods for trajectory optimization. Annals of Operations Research 37 (1992) 357--373.

....k ; t k ) 8f(y k2 ; u k2 ; t k2 ) Gamma f(y k 1 ; u k 1 ; t k 1 ) Gamma y k2 = 0; 4. 8) with u k2 = 1 2 (u k u k 1 ) 1 8 Deltat(w k Gamma w k 1 ) A similar scheme using a 2 stage Lobatto IRK is proposed by Betts and Huffman [5] See also, Pesch [28] Lamour [24] von Stryk and Bulirsch [36], Bulirsch et al. 8] von Stryk [35] Betts [3] and Schulz, Bock and Steinbach [32] An important property of collocation is that the partial derivatives of problem functions are relatively simple to compute. For the moment, suppose that the relation u k2 = 1 2 (u k u k 1 ) 1 8 Deltat(w ....

O. von Stryk and R. Bulirsch, Direct and indirect methods for trajectory optimization, Ann. Oper. Res., 37 (1992), pp. 357--373.

....when a state or control variable inequality constraint (3) becomes active or inactive during a phase. Numerical optimal control methods based on the Euler Lagrange differential equations (EL DEQs) and the Maximum Principle (MP) can mainly be divided into two classes: direct and indirect methods [16]. Indirect methods approximate a solution by explicitly solving first and second order optimality conditions resulting from EL DEQs and the MP. For reasons already discussed in [6, 12, 16] they are not flexible enough for the purpose needed here. Direct methods are based on a transcription of ....

....(EL DEQs) and the Maximum Principle (MP) can mainly be divided into two classes: direct and indirect methods [16] Indirect methods approximate a solution by explicitly solving first and second order optimality conditions resulting from EL DEQs and the MP. For reasons already discussed in [6, 12, 16] they are not flexible enough for the purpose needed here. Direct methods are based on a transcription of optimal control problems into (finite dimensional) nonlinearly constrained optimization problems (NLPs) either by direct shooting or direct collocation [5, 16] Direct methods promise high ....

[Article contains additional citation context not shown here]

O. von Stryk, R. Bulirsch. Direct and indirect methods for trajectory optimization. Annals of Operations Research 37:357--373, 1992.

No context found.

von Stryk, O. and R. Bulirsch, `Direct and indirect methods for trajectory optimization', Annals of Operations Research, 37, 357--373 (1992).

....are also parameterized or, in other words, if they are also chosen from a finite dimensional space. The equations of motion are then satisfied only pointwise by prescribing so called collocation conditions. A description of methods belonging to this class can be found, e.g. in [31] 16] [34], 33] and [19] Among the indirect methods, the multiple shooting method has several advantages, for example, its outstanding accuracy and the possibility to verify many necessary conditions. In addition, inequality constraints and interior point constraints can be treated, too, and, which is of ....

....is qualified for an application on vector or parallel computers (see [21] Among the direct methods, direct collocation has the advantage that no explicit integration must be carried through. Thus, this method is expected to be efficient. Recently, a so called hybrid approach was suggested (see [34] and [33] where just those two methods, direct collocation and multiple shooting, are combined in the following way: The numerical approximation of the adjoint variables of the Lagrangian of the associated nonlinear programming problem is used to approximate the adjoint variables of the optimal ....

[Article contains additional citation context not shown here]

von Stryk, O. and Bulirsch, R.: Direct and Indirect Methods for Trajectory Optimization, Annals of Operations Research 37, 357--373, 1992.

....available robot, discuss several objectives for optimal trajectories and consider state constraints on the angular velocities that play an important role in the time optimal motion. In our approach, we combine a direct collocation and an indirect multiple shooting method in an hybrid approach (cf. [28]) with a large domain of convergence and highly accurate solutions. The direct collocation method is easily capable to treat a wide variety of objectives and constraints on the state and control variables. 2 Oskar von Stryk and Maximilian Schlemmer 2 Problem Statement Figure 1: Three degrees of ....

....collocation approach is a finite dimensional approximation of control and state variables, i.e. a discretization. Here, we choose a continuous, piecewise linear control approximation and a continuously differentiable, piecewise cubic state approximation, cf. Hargraves, Paris [11] and [25] 26] [28]. The differential equations, the state and control constraints are only pointwise fulfilled in this approach. The discretization results in a nonlinear optimization problem subject to nonlinear constraints. Convergence properties of the method and details of an efficient implementation are ....

[Article contains additional citation context not shown here]

von Stryk, O., Bulirsch, R. Direct and indirect methods for trajectory optimization. Annals of Operations Research 37 (1992) 357-373.

No context found.

Stryk, O. v. and Bulirsch, R., "Direct and indirect methods for trajectory optimization," Annals of Operations Research, Vol. 37, 1992, pp. 357--373.

No context found.

von Stryk, O., and R. Bulirsch, `Direct and Indirect Methods for Trajectory Optimization ', Annals of Operations Research, 37, 357--373 (1992).

No context found.

von Stryk, O. and Bulirsch, R., Direct and Indirect Methods for Trajectory Optimization, Annals of Operations Research, to appear.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC