1998 - Draper Laboratory

where x is the downrange position (relative to the target), y is thecrossrange position, z is the target-relative altitude, u is thedownrange velocity, v is the crossrange velocity, w is the verticalvelocity, a x is the acceleration in the horizontal direction, a y is theacceleration in the crossrange direction, a z is the acceleration inthe vertical direction, and g is the local gravitational acceleration.The Performance Index andProblem SetupThe minimum control effort (acceleration) problem can be posed as1 t f 2 2 2min J = Γt f + ∫ a x + a a dty + z τ2 o(7)subject to the aforementioned dynamics (Eqs. (1)-(6)), withspecified initial conditionsx0 y0 = xV o− xT= yV o− yT(8)(9)z0 = zV o−zT(10)u0 v0 w0 ===uVovVowVo(11)(12)(13)and the following terminal constraints[xf yf zf u f vf wf (14)Γ is a weighting parameter on the final time (if it is unspecified).This represents a trade-off between the minimum time problemand the minimum control effort problem. It can be set to zero ifdesired. The Hamiltonian and the Bolza function for thisproblem, respectively, areH a 2 2a 2x y a z= + + + λxu+ λyv+ λzw+ λuax + λvay+ λ w( az+ g)2 2 2(15)G = Γt f + υxxf + υyyf + υzz f + υuu f + υvvf + υwwfThe First VariationConditions(16)The Euler-Lagrange equations, along with the transversalitycondition on the final state, yield the following equations (Refs.[3], [4])˙λ x˙λ y==00(17)(18)˙λ˙λ˙λuvw˙λ z = 0= −λx= − λy= − λz(19)(20)(21)(22)with the terminal conditionsIf we definethe Lagrange multipliers are found to beand the control variables, a x and a z areThe states can be written asax = − λu = −υxtgo−υua y = − λv = −υ ytgo−υvaz = − λw = −υztgo−υwz3 2 2υztgo υw tgo gtgo= − − +6 2 2wλλλλλλyfzfufvfwftgo∆ t f −tλλλx fxyz=========υυυυυυυυυλ = υ t + υu x go uλ = υ t + υv y go vλ = υ t + υw z go w3 2υxtgo υutgox = − −6 23 2υ ytgo υvtgoy = − −6 22υ xtgou = + υ utgo22υ ytgov = + υ vtgo22υ ztgo= + υwtgo− gtgo2xyzxyzuvw(23)(24)(25)(26)(27)(28)(29)(30)(31)(32)(33)(34)(35)(36)(37)(38)(39)(40)(41)(42)(43)(44)An Optimal Guidance Law for Planetary Landing3