COMPARISON OF DIFFERENT LATERAL ACCELERATION AUTOPILOTS FOR A SURFACE-TO-SURFACE MISSILE

This paper presents a comparison of three lateral acceleration autopilots for a surface-to-surface missile: three-loop conventional acceleration autopilot, and gamma-dot and three-loop acceleration autopilot based upon the inverse-dynamic control. The surface-tosurface missile motion is described by nonlinear differential equations whose parameters change rapidly over a very wide range due to variable velocity and altitude. The requirement for the accurate controlling of the missile in such an environment represents a challenge for the autopilot designer. The brief review of the calculation of the autopilot gains is given using the concept of the “point” stability for the linear time-varying system with “frozen” dynamic coefficients. The method of the inverse-dynamic control is presented in the next section for two types of the autopilots: gamma-dot and acceleration autopilot. Both of them require the design of the estimators for the variables used as inputs to the control law. Finally, six-degree-of-freedom simulation results of the missile response to the demanded command on the typical ballistic trajectory are presented. The comparison of three autopilots considers the steady state errors and the sensitivity of the response to the highly variable environment. It was shown that the inverse-dynamic control can be very effective in the controlling of the surface-to-surface missile. UDC: 623.462.2 ; 629.7.051.5 FIELD: Mechanical engineering (Rocket technique)

Introduction he aim of this paper is to present the main results of the comparison of different autopilot designs for a surface-to-surface missile (SSM).The free flight rocket designed to achieve the range about 50 km was modified to a guided weapon with reduced dispersion.The missile carries the flight computer, inertial measurement unit (IMU) with three rate gyros and three accelerometers, autopilot, and control section.The free flight rolling rocket was redesigned to canard a configuration whose control and guidance sections were stabilized against the rolling motion due to the application of the IMU and the strapdown navigation algorithm.
The purpose of the autopilot is to control the modified SSM during the whole flight, i.e., at low-, medium-and high altitude flight conditions.This is a difficult problem because the aerodynamic controls are very sensitive to the high variation of the air density, Mach number, angle of attack, and other missile dynamic properties needed for the autopilot design.
Typically, the design of the autopilot for the missile is based on the concept of the point stability technique.Most of the autopilots have a fixed structure whose gains are scheduled upon flight conditions such as dynamic pressure and the angle of attack.Therefore, tedious process of generating numerous aerodynamic transfer functions is required.After this step, the "frozen-point" stability is done to develop autopilot gains for the purpose of gain scheduling.This method of the autopilot design was described in many references such as Refs.[1], [2] and [3].The authors of this paper have developed the computer codes for the complete numerical linearization of six-degree-of-freedom (6-DOF) model [4] and the autopilot design [5].Different reference trajectories (ballistic, constant manoeuvre, straight line) were employed to generate aerodynamic transfer functions.One of the approaches in preparing the linear missile model based upon 6-DOF concept is shown in Ref. [6].
Many modern control methods with the application to the autopilot designs were published in papers, Refs.[7] - [10].The method of the robust nonlinear inverse-dynamic control, or feedback linearization, [11] solves the problem of the synthesis of the autopilot for the missile having a highly varying nature.In addition to large variations in aerodynamics, mass, and inertia properties of a SSM, which occur during the boost phase, several constraints were faced by the autopilot designers.The problem of a small manoeuvre capability at high altitudes can be solved partially by allowance of small static instability (the centre of mass is behind T the centre of pressure).The Mach number is changed during the boost phase from the zero value to that greater than 3, when the static instability appears.The dynamic instability and dispersion for a free-flight rocket with extended range was analyzed in details in Ref. [12].The robust control using the inverse-dynamic method can be applied to the SSM independently of the time instant on the trajectory even in the case of the dynamic instability [13].
In this paper, the results of different autopilot designs are discussed.The three-loop conventional acceleration autopilot is studied first.Using the concept of point stability, all necessary expressions for the autopilot's gains are rewritten from the published papers.The next section includes the method of the inverse-dynamic control and its application to two types of autopilots: gamma-dot and three-loop lateral acceleration autopilots.The numerical simulation results using a 6-DOF model will be presented to compare the SSM responses to the given commands on the trajectory for all three autopilot designs.Finally, some conclusions are given at the end of this paper.
The autopilot designs, presented in this paper, can be applied to both the missiles with Lambert guidance [14] and those without a thrust terminating mechanism [15].

Three-loop Conventional Acceleration Autopilot
Classical control techniques have dominated missile autopilot designs over the past several decades.Most missile autopilots employ acceleration and rate feedback with the proportional and integral control to stabilize the statically unstable missile and to track the guidance commands (Fig. 1).
The aerodynamic transfer functions of the pitch rate and normal acceleration to the controls deflections in Fig. 1 are, respectively ( ) ( ) ( ) where ( ) k U V ≈ -missile velocity, w z , -derivatives of linear acceleration with respect to the parameter in the subscript, w m , q m , m η -derivatives of angular acceleration with respect to the parameter in the subscript, Ref. [16].If the reference trajectory is ballistic, q q Δ = ,  ς should be modified as it was shown in [1] and [3].From Fig. 1, the overall transfer function of the lateral acceleration to demanded acceleration can be developed as: where ( ) The gains of the autopilot , ,  Equating the corresponding coefficients in the denumerators of Eqs. ( 12) and (7) gives the following expressions for the autopilot gains: ( ) In a general case, the steady state gain 1 ≠ .Therefore, the pre-gain ac k K = may be required to achieve the demanded acceleration.

Theoretical Background
This section follows the description of the inverse-dynamic control given in Ref. [13].The missile's nonlinear equations of motion can be separated into a nonlinear homogeneous term plus the term which is the linear function of the control vector where x is a 1 n × state vector and u is the control vector of the dimension × matrix of control sensitivity.We assume that the 1 m × output vector y is a linear function of the state vector = y Hx The matrix H is selected by the designer.The derivative of the chosen output vector is: the inverting control law is obtained The control law includes a model of missile's homogeneous dynamics in the feedback loop and the inverse of its control effects in the forward loop (Fig. 2).The final step is to design a controller for the system described by Eq. ( 19).The matrix ( ) ( ) = * G x HG x is often not invertible because some elements of y are not linear functions of u .The repeated differentiation eventually reveals a linear relationship.Suppose that this can be applied to the first element of y , i.e., 1 y where 1 h is the first row of H .The second derivative of 1 y is If the necessary linear relationship between u and 1 y && is established, the first rows of ( ) , respectively.
The second element of y is checked then.If Eq. ( 22) does not produce a linear relationship, the next derivative of 1 y is generated.We denote the vector of suitable derivatives of y by

Gamma -Dot Autopilot
The gamma-dot autopilot is used to control the velocity vector turn rate, γ& .It is useful in the case of SSM with proportional navigation because it provides explicit control of the missile velocity vector turn rate required by the guidance law.If the missile has the pitch and yaw rate gyros for measuring body rates, an estimator is implemented to reconstruct the missile gamma-dots.The error in estimating the gamma-dots depends on the errors in estimating missile's aerodynamic characteristics.In the ca-se of full strapdown navigation systems, the gamma-dots and all the other parameters required by the autopilot are generated using a navigation algorithm.Since the pitch and yaw channels are identical, only the former will be described.
Using notations in Fig. 3, we can develop a simplified model for missile's motion.The velocity vector turn rate is given by where Z , c Z x F , X , m are normal force due to angle-of-attack and controls deflection, thrust, aerodynamic axial force, and missile mass, respectively.
into Eq.( 24), we get where , , , , We select the following state variables: Using the kinematical relationship q after differentiating Eqs. ( 29) and (30), we get ( ) Now, we can implement the method of the inverse-dynamic control.
The estimated quantity γ& will be a controlled variable.After an additional differentiating step in Eq. ( 32) and introducing the new state variables we get the transformed system of differential equations where The new system described by Eqs.(36) and (37) is a linear, timeinvariant system with a new control input v .The deflection of control surfaces may be easily determined from Eq. (38) in terms of the new control input v and the parameters of missile motion.
The control law for the new time-invariant system can be defined to satisfy some design requirements where c γ& is the demanded value of the velocity vector turn rate and Equation ( 38) can be solved for the deflection of controls as where The block-diagram of the gamma-dot autopilot is shown in Fig. 4. The deflection of controls depends on the measured value of the pitch rate q , estimated values of ˆ, γ α & , and the additional correction ( ) cor t η which is the consequence of the variable velocity vector turn rate derivative ( ) , the correction to the deflection of controls is equal to zero.

Three-Loop Acceleration Autopilot
The inverse-dynamic control will be applied to the three-loop acceleration autopilot as well.Instead of using the velocity turn rate γ& as a state variable, the normal acceleration is introduced Since the second state variable remains the pitch rate ( 2 x q = ), the simplified state space model takes the following form: The controlled variable is normal acceleration, 1 n y a = .Taking two differentiations on the normal acceleration and defining new variables in the transformed system, 1 2 , n n y a y a = = & , we get again: Hence, we get the time-invariant system and the design of the optimal autopilot can be done easily.Again, after two differentiations the linear function with respect to the deflection of controls in the expression for a new control input v , Eq. ( 52), is obtained.
The integral control law was applied ( ) The transfer function of the normal acceleration 1 n y a = to the demanded value of the normal acceleration nc a follows from Eqs. ( 50), ( 51) and (53): The D.C. gain is equal to unity, and the zero steady-state error occurs.From Eqs. ( 50) and (51), with the new controlled variable given by Eq. ( 53), the solution of Eq. ( 52) for the controls deflection η gives: ( ) () where The control law of the three-loop acceleration autopilot generated by the inverse-dynamic control is similar to that of the gamma-dot autopilot.In order to obtain a linear time-invariant system, instead of the pitch angle feedback as in the case of the conventional acceleration autopilot, we need the angle-of-attack feedback [see Eq. ( 55)].The autopilot is desig-ned by using the estimator for the angle-of-attack.The compensation of the rapid changes in g α is introduced via the corrected value of the con-

Numerical Examples and Simulation
To determine the response of the missile having different autopilot designs, the six-degree-of-freedom model and numerical simulation were applied.The missile configuration used in the simulation study is an artillery rocket of the range about 50 km.The aspect ratio of the body is / 17,8 l d = . The canard section is built into the nose section inside the diameter of the cylindrical body at the distance of / 1,55 l d = from the missile tip.The fin-stabilized configuration has four pop-out fins without cant for generating the rolling motion.The roll autopilot is used to stabilize the canard section with the IMU against the rolling motion.The rocket motor having the total impulse of 33500 dNs imparts the velocity over Mach 3 to the missile to achieve the range over 50 km.The motor has two levels of the thrust 27800/8000 dN with burn times of 0.17/3.80sec.The missile weight, the mass centre location from the missile tip, the roll and the transverse inertia radius before and after burn is 390/225 kg, 2.65/2.47m, 0.094/0.098and 1.2/1.4m, respectively.The elevation angle of 54.19 deg was chosen to achieve the range of 50 km.The apogee of the trajectory is 19600 m.The aerodynamic data for the 6-DOF model were generated by numerical simulation and wind tunnel tests.
The missile configuration is statically unstable in the time interval (3,9;5,1) s t ∈ when a high value of the Mach number is achieved.Two control points were selected for the autopilot design: 10 s t = (after the burn out time when the missile becomes statically stable again), and 60 s t = (near to the apogee of the trajectory).Using the "frozen'' aerodynamic transfer functions parameters for 10 s t = and the desired autopilot's dynamics given by The responses of the autopilot to the demanded unit step input for 3,5,10, and 15 s t = are shown in Fig. 5.The time varying transfer functions parameters have important effects on the autopilot's responses which are characterized by an increased dynamic errors if the time is different from the point selected for the design.The response of the missile to the given command of 1 g ± by using the 6-DOF model for the gains in Eq. ( 61) is shown in Figs. 6 and 7 for the normal acceleration and the deflection of controls and angle-of-attack, respectively.A more realistic numerical simulation by the 6-DOF model shows that the satisfactory autopilot response was not obtain even in the vicinity of the chosen control point, 10 s t = . The additional adjustment of the autopilot is required.The change of the gain ac K to the value of 0.02 ac K = gives the response with the reduced steady state error, but with the increased overshoot, as shown in Fig. 8.The concept of the "frozen" point stability in the design of the three-loop conventional acceleration autopilot cannot give good results for the wide interval of the missile flight.
The identical procedure was repeated for the control point near to the apogee of the trajectory, 60 s t = . Since the natural frequency of the missile is reduced to , the desired values of the autopilot dynamics were chosen to be 6 rad/s, 0,5 s, 0,6 . The new values of the autopilot gains were obtained as 0,057 s/m, . The results of the numericalsimulation by the 6-DOF model in Fig. 9 show again that the additional adjustments of the gains are useful.Since the manoeuvre capability of the SSM is highly reduced at the apogee, the demanded acceleration was 0.1 g ± .
The gamma-dot autopilot was designed using the inverse dynamic control for the same desired dynamic characteristics as those for a conventional autopilot at t=10 s, Eq. ( 60).Based on these values, the coefficients in the transfer function Eq. (40) were calculated as  42) -(45) were used for the calculation of the autopilot gains and the correction of the controls deflection due to time varying dynamics needed for the control law, defined by Eq. (41).Fig. 10 illustrates the autopilot response to a step command 1 deg/s ± .
In comparison to the conventional autopilot, the gamma-dot autopilot shows better performances because the desired dynamics was achieved without steady state errors.For the same desired dynamics as in the case of the conventional autopilot at t=60 s, the response to the demanded value of the velocity vector turn rate of 0,1 deg/s ± is shown in Fig. 11.The comparison of the diagrams in Figs. 9 and 11 proves some advantages of the gamma-dot over the conventional autopilot: faster response and lower values of dynamic errors in tracking the demanded velocity vector turn rate.

Figure 1 -
Figure 1 -Three-loop conventional acceleration autopilot It is assumed that the acceleration is at the mass centre (CM), z z a a ′ Δ = Δ .If the accelerometer is away from the CM, the transfer function

Figure 2 -
Figure 2 -Inverse dynamic control invertible for linearly independent controls.If v represents the desired values of ( ) d y , the inverting control is described by Eq. (20).The application of the inverse control can be complex because the evaluation of ( ) that a full, d − differentiable model of the missile be included in the control system.It is worth to say that the inverse-dynamic control can be applied to both nonlinear and linear time-varying systems.

Figure 3 -
Figure 3 -Forces and moments acting on the missile Substituting the following approximations , , , , , sin , cos 1, and the time constant τ may be defined through the synthesis of the autopilot.The application of Laplace transformation in Eqs.(36), (37) and (39) and solving for 1 y γ = & in terms of demanded input c γ& gives the transfer function of the overall timeinvariant system the placement of the closed-loop poles, while the time constant τ is selected to compensate for the aperiodic time lag in the autopilot loop.

Figure 5 -
Figure 5 -Responses of a conventional autopilot designed for t=10 s

Figure 12 -Figure 13 -
Figure 12 -Three-loop acceleration autopilot designed by the inverse-dynamic control ( 10 s t = ) be chosen in such a way to provide the desired parameters of the closed loop e