Numerical solution of optimal control problems with state constraints by sequential quadratic programming in function space

Abstract

Control constraints, i.e. the control u must satisfy : S 0 (u(t),t)~ 0 for all 0~ t ~ T. (1.1 .5)* Mixed control state constraints, i.e. the control u and the state x must satisfy: for all 0~ t ~ T. (1.1 .6)* State constraints, i.e. the state x must satisfy : Si(x(t ),t) ~ 0 for all 0~ t ~ T. ( 1.1.7)For the numerical method to be presented in this book the distinction between control and mixed control state constraints is not important.The distinction between mixed control state constraints and state constraints however, is essential.The major difficulty involved with state constraints is that these constraints represent implicit constraints on the control.as the state function is completely determined by the control via the differential equations.The optimal control problems formally stated above are obviously of a very general type and cover a large number of problems considered by the available optimal control theory.The first practical applications of optimal control theory were in the field of aero-space engineering, which involved mainly problems of flight path optimization of airplanes and space vehicles.(See e.g.Falb et al. (1966Falb et al. ( , 1969)), Bryson et al. (1975).)As examples of these types of problems one may consider the problems solved in Sections 8.1 and 8.2.We note that the reentry manoever of an Apollo capsule was first posed as an optimal control problem as early as 1963 by Bryson et al. (1963b).Later optimal control theory found application in many other areas of applied science.such as econometrics (see e.g.van Loon (1982), Geerts (1985)).Recently, there is a growing interest in optimal control theory arising from the field of robotics (see e.g.Bobrow et al. (1985), Bryson et al. (1985).Gomez (1985), Machielsen (1983).Newman et al. (1986). Shin et al. (1985)).For the practical application of the method presented in this tract this area of robotics is of special importance.Therefore we will briefly outline an important problem from this field in the next section.1.2.An example of state constrained optimal control problems in robotics.In general.a (rigid body) model of a robotic arm mechanism.which consists of k links (and joints) may be described by means of a nonlinearly coupled set of k -differential equations of the form (see e.g.Paul (1981).Machielsen (1983)):where q is the vector of joint positions.cj is the vector of joint velocities and if is the vector of joint accelerations.J (q) is the k xk inertia matrix which. in general.will be invertible.The vector D (cj .q ) represents gravity.coriolis and centripetal forces.F is the vector of joint torques.It is supposed that the arm mechanism is to be controlled from one point to another point along a path that is specified as a parameterized curve.The curve is assumed to be given by a set of k functions Y; :[0,1]-> JR of a single parameters, so that the joint positions q; (t) must satisfy : minimize f (x ).