Five-Degrees-of-Freedom Simulation

Frequently, three-DoF models, as described in the preceding chapter, do not model in sufficient detail the vehicle dynamics. Hence we may add two attitude degrees of freedom to the three translational equations and call the composite a five-DoF simulation. For a vehicle that executes skid-to-turn maneuvers (an inter­cept missile), pitch and yaw attitude dynamics are incorporated. For a bank-to-turn aircraft, the yaw angle of the missile is replaced by the bank angle. Euler’s law for­mulates the differential equations for the two attitude angles. However, the increase in complexity is significant and approaches that of a full six-DoF simulation. To maintain the simple features of a three-DoF simulation and at the same time account for the attitude dynamics, the transfer functions of the closed-loop autopilot replace Euler’s equations. This implementation is called a pseudo-five-DoF simulation. The word pseudo conveys the meaning of approximating the attitude dynamics with the linear differential equations of the transfer functions.

Pseudo-five-DoF simulations are popular models for concepts that are only loosely defined. During preliminary design, the vehicle’s aerodynamics may be sketchy, the autopilot design rudimentary, and the guidance and navigation im­plementations uncertain. These are good reasons to match these notional systems with the simple pseudo-five-DoF models. If you want to find out whether a simula­tion has this pseudo characteristic, look for these telltales: trimmed aerodynamics, angle-of-attack as the output from a transfer function, body rates not obtained by solving the Euler’s equations, and the absence of controls and actuator models.

Using the CADAC environment (see Appendix В), I have built such simulations for medium range air-to-air missiles, air-to-ground guided bombs, cruise missiles, airplanes, antisatellite interceptors, and reentry vehicles. These simulations were in support of either concept evaluations or man-in-the-loop simulators. It is amazing how useful these bare-bones models are. They make trade studies feasible, yield quick results for those hurried marketers, and are easily modified for other applica­tions. One feature is particularly important: the integration step can be one or even two orders of magnitude greater than that of a six-DoF simulation. When execution time is critical as in air combat simulators, these pseudo-five-DoF models may be the only feasible approach. What enables the greater time steps is the disregard of high-frequency phenomena, like attitude motions, fast autopilots, actuators, and sensor dynamics.

Some modelers are more ambitious and would like to create a six-DoF show­piece. They add the rolling transfer function of missiles or the yawing transfer function of aircraft to the dynamics and thus create a pseudo-six-DoF simulation. This expansion is easily accomplished and may be beneficial when the attitude dynamics are emphasized. However, the pseudo limitations still apply, and it is doubtful that much fidelity is gained without the modeling of controls and higher – order dynamics.

Finally, a pseudo-five – or six-DoF simulation can become the trailblazer for the full six-DoF masterpiece. The aerodynamics is replaced by untrimmed data including aerodynamic moments and control effectiveness. Euler’s equations are introduced to solve the three attitude degrees of freedom, and autopilot details and actuator dynamics increase model fidelity. If your pseudo-five-DoF had a complete guidance loop, you may be able to transfer it directly. I took that shortcut for several air-to-air missile simulations. The sensor and guidance algorithms developed ear­lier during the conceptual phase worked perfectly well in the six-DoF simulation.

In this chapter we will concentrate on the pseudo-five-DoF simulations for ro­tating round Earth (strategic missiles, hypersonic aircraft, and orbital vehicles) and for flat Earth (tactical missile and aircraft applications). The equations of motion are based on Newton’s second law and supplemented by kinematic equations that calculate the attitude angles. If you need the sixth pseudo-DoF, you should be able to add it yourself. On the other hand, if you want to develop a full five-DoF simulation you should turn to Chapter 10, and reduce your model by one degree of freedom.

My plan is to derive the equations of motion in tensor form, provide the relevant coordinate transformations, and express them in matrix form for programming. The right-hand sides of these equations consist of the externally applied forces. We will develop these forces from the inside out, beginning with the trimmed aero­dynamics for missiles or aircraft, the propulsive forces of rockets or turbojets, and the gravitational acceleration. Then we enlarge the circle and discuss how autopi­lots control these aerodynamic forces through acceleration and altitude commands for both skid-to-tum and bank-to-tum vehicles. Finally, the guidance law places demands on the autopilot to achieve certain trajectory objectives. We will discuss proportional navigation for target intercept and line guidance for trajectory shaping (waypoint guidance and automatic landing approaches). We conclude by address­ing electro-optical or microwave sensors that provide the target line of sight to the guidance processor.

The CADAC CD provides several examples of pseudo-five-DoF simulations. Besides the simple and more complex air-to-air missile simulations AIM5 and SRAAM5, you can find a generic cruise missile CRUISE5. With the material covered in this chapter, you should be able to decipher their source code, make some test runs, and adapt them to your own needs.