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CHAPTER 2 REFERENCE FRAMES xm 050 050 -075 -075 ym -025 025 025 -025 zm 000 000 000 000
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Assuming that the o set from the tangent frame origin to the vehicle frame origin is [100, 50, 7] m, for each of the following attitudes, compute the tangent plane location for each corner of the vehicle Attitude, deg 4500 000 000 000 4500 -3000 4500 4500 000 4500 -4500 000 Exercise 29 Use direct multiplication to show that to rst order, the product of (I a ) and (I + a ) is the identity matrix ba ba Exercise 210 Use eqn (B15) to form the skew symmetric matrix By direct matrix multiplication, con rm eqns (265 268) Exercise 211 Section 272 derived the relationship between the Euler attitude rates = ( , , ) and the body-frame inertial angular rate vector b ib 1 Use a similar approach to show that the relationship between the Euler attitude rates and the angular rate of the body-frame relative to the inertial-frame represented in geodetic frame is g = T ib where cos( ) cos( ) sin( ) 0 cos( ) 0 (280) T = sin( ) cos( ) sin( ) 0 1 2 Check the above result by con rming that Rg = T 1 E b This matrix T is useful for relating the attitude error represented in geodetic frame (de ned in Section 105) to the Euler angle errors = ( , , ) The relation = T is used in Sections 1055 and 1251 Exercise 212 Figure 91 shows the axle of a vehicle Denote the length of the axle by L The body frame velocity and angular rate are v = [u, v, 0] and = [0, 0, 1] Use the law of Coriolis to nd the tangent plane velocity of each wheel
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The quantitative analysis of navigation systems will require analytic system models Models can take a variety of forms For nite dimensional linear systems with zero initial conditions that evolve in continuous-time, for example, the ordinary di erential equation, transfer function, and state space models are equivalent The dynamics of the physical systems of interest in navigation applications typically evolve in continuous-time, while the equations of the navigation system itself are often most e ciently implemented in discrete-time Therefore, both di erence and di erential equations are of interest This chapter present several essential concepts from linear and nonlinear systems theory
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Three equivalent model structures for continuous-time systems are discussed in this section The models are equivalent in the sense that each contains the same basic information about the system However, some forms of analysis are more convenient in one model format than in another Models for physical systems derived from basic principles often result in ordinary di erential equation or state space models Frequency response analysis and frequency domain system identi cation techniques utilize the transfer function representation Optimal state estimation techniques are most conveniently presented and implemented using the state space approach A major objective of this section is to provide the means to translate e ciently and accurately between these three model representations 63
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CHAPTER 3 DETERMINISTIC SYSTEMS
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Ordinary Di erential Equations
Many systems that evolve dynamically as a function of a continuous-time variable can be modeled e ectively by a set of n-th order ordinary di erential equations (ODE) When the applicable equations are linear, each n-th order di erential equation is represented as y (n) (t) + d1 (t)y (n 1) (t) + + dn (t)y(t) = n1 (t)u(n 1) (t) + + nn (t)u(t)
(31)
where the notation ()(j) denotes the j-th time derivative of the term in parenthesis In this general form, the coe cients of the di erential equation are time-varying In applications involving time-invariant systems, the coe cients are constants Eqn (31) represents a single-input single-output dynamical system The input signal is represented by u(t) The output signal is represented by y(t) Eqn (31) is referred to as the input-output ordinary di erential equation When the applicable equations are nonlinear, the n-th order di erential equation is represented in general as y (n) (t) = f (y (n 1) (t), , y(t), u(n 1) (t), , u(t)) (32)
Taylor series analysis of eqn (32) about a nominal trajectory can be used to provide a linear model described as in eqn (31) for local analysis See Section 33 To solve an n-th order di erential equation for t t0 requires n pieces of information (eg, initial conditions) which describe the state of the system at time t0 This concept of system state will be made concrete in Section 35 Example 31 Consider a one dimensional frictionless system corresponding to an object with a known external force applied at the center of gravity The corresponding di erential equation is m (t) = f (t) p (33)
where m is the mass, p(t) is the acceleration, and f (t) is the external applied force If the object is also subject to linear friction and restoring forces, then this is the classic forced mass-spring-damper example, The resulting di erential equation is m (t) + bp(t) + kp(t) = f (t) p (34)
where b represents the linear coe cient of friction and k is the coe cient of the linear restoring force
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