Guidance and Navigation Systems CHAPTER 2: GUIDANCE


 Brenda Knight
 1 years ago
 Views:
Transcription
1 Guidance and Navigation Systems CHAPTER : GUIDANCE Guidance as closed loop system Guidance Classification Proportional Navigation Guidance (PNG) Optimal Guidance Other Guidance Schemes Advanced Guidance Waypoint Guidance Noise and Uncertainties Examples and Applications.1. Guidance as closed loop system The most important feature of guidance, at least in this context, is the tight association with the autopilot and the inner control loop. Therefore, unless otherwise specified, we will address only those topics that involve some form of feedback. As described in the introduction, the guidance process is primarily a kinematic process and has several theoretical and computational components. Measurement of vehicle position and velocity, Assessment of target dynamic behavior (final point or intermediate points), Correction and adjustment to external noise and disturbances, Computation of control actions necessary to properly adjust position and velocity, Delivery of suitable adjustment commands to the vehicle's control system. Let us review the guidance block diagram from chapter 1. 1
2 The first block that we can identify is the inner control loop, which includes the complete dynamic model of the vehicle, actuators, external disturbances, and flight control system (inclusive of autopilot). The resulting command of interest, from a guidance stand point, is the acceleration, here a lateral acceleration A my. This loop, as we can see from the input going into the flight control system, is responsible for disturbance rejection, rotational motion control, and translational motion control in the direction of the vehicle s acceleration vector. The dynamic behavior that relates the motion of the vehicle with the motion of the target point is shown above. The objective in this particular case is the management of the relative motion of the two systems. The target dynamics may be known or unknown, and the output of this process is related to the relative instantaneous distance between the two, indicated by the term 1/R. The next block shows how the information relative to distance and direction is measured by the vehicle (through a component called seeker ), processed with noise models based on the component, target, and environment knowledge, and finally sent to the guidance process. In this particular application, the main variable used by the guidance law is the socalled line of sight rate or LOSrate ( σ& ). Of course this is only one possible implementation (applicable to high performance, manoeuvrable vehicles), and others are possible.
3 The next block diagram describes the details of the main interconnections among the guidance processor, seeker, and angular measurement unit (navigation system). The actual guidance system is shown below, from a process point of view, meaning that that guidance method and structure is not detailed here. As we can see, the input LOS rate is smoothed and filtered. In addition vehicle s attitude rate is fed in and command to the flight control system is given. An interesting element is the saturation block, which is present due to limitations in the actuators, and introduces stability problems. Another example of the closed loop nature of guidance is shown next, taken from a marine rather than aerospace application. The guidance process is defined using the concept of waypoints (more on this later). In very general terms, a surface ship block diagram is shown below: 3
4 As we can see, the guidance system is responsible for providing desired position and rate variables [ ν d, η d] to be implemented by the control system (or autopilot). The loop is closed by the navigation system responsible for locating the ship, reference command and information about disturbances. A different look at the same problem is shown next. For a ship or an underwater vehicle, the guidance and control system usually consists of: an attitude control system a path control system In its simplest form the attitude control system is a course autopilot. The main function of the attitude feedback control system is to maintain the vessel in the desired attitude on the ordered path by controlling the vessel in roll, pitch, and yaw. The task of the path controller is to keep the vessel on the predefined path with some predefined dynamics (e.g. forward speed) by generating orders to the attitude control system. The waypoints are stored in a waypoint database and used for generation of a trajectory or a path for the moving vessel to follow. Both trajectory and maneuvering control systems can be designed for this purpose. Sophisticated features like weather routing, obstacle avoidance and mission planning can be incorporated in the design of waypoint guidance systems... Guidance Classification There are many implementations of guidance, according to the type of vehicle, type of application, required performance, sensors and actuators, etc. Therefore many different classifications are possible. 4
5 ..1. Trajectory Phases Classification One way of separating guidance design and requirements is as a function of the part of the mission. In the case of a vehicle or a rocket, for instance, there are three phases: initial, midcourse, and terminal. The initial phase is characterized by a series of tasks, some of them extremely demanding from the standpoint of the control system, since the dynamic behavior is highly nonlinear and uncertain. In terms of guidance we can recognize: Launcher clearance Initial acceleration Deployment of flight surfaces Inertial navigation initialization Alignment Error In the midcourse phase, guidance solutions may accommodate open loop strategies with preset corrections: Cruise course/speed Closure on the target Flight Control System & Navigation Control System corrections Onboard Sensors External Tracking The third and last phase is the terminal part of the task. In this phase, the dynamics depend heavily on the relative motion with the target, if the target is not stationary. The guidance system may rely on onboard processing of imagery and radar information, in order to achieve the objective. This phase is the most challenging from the point of view of control system theory and application. Target sensing Vehicle performance Violent, high Gforce maneuvers 5
6 Target Intercept The three phases of flight can be also identified in a completely different mission. Let us consider the mission of traveling to Mars (courtesy of NASAJPL). It is obviously a complex mission with different engineering areas of interest. We consider here the mission of the Global Surveyor spacecraft (second from the top). The main scientific tasks of the spacecraft are relative to mapping of the surface, analysis of the atmosphere, and relay for future missions. The general timeline is shown above. The initial phase is given by the launch, parking orbit, preparation for Mars transfer; the midcourse phase, which is the longest; the final phase that includes all operations about Mars. The initial phase starts with the launch and all preset operations to reach Earth orbit. Based on the current astronomical configuration of Earth, Sun and Mars, and the timing of arrival, the launch is set, the rocket rolls to the Earth orbit osculating plane and ends its operation, once reached a given altitude and velocity capable of inserting the spacecraft in a parking orbit. The guidance system in this phase is essentially coincident with the trajectory control system of the launcher. The midcourse phase consists of three parts. The first is the departure from Earth s gravity field, the second is the insertion into the transfer trajectory to Mars, and the third is the deceleration into Mars orbit. The approximate technique used for this type of motion is based on the method of patchedconics, which connects segments of 6
7 Keplerian orbits, to come up with the midcourse phase. The guidance system is strongly connected to the navigation system since precise position and timing are crucial. The implementation of guidance commands is usually preset, unless major position errors are found along the flight. Earth departure is obtained by performing a + V change, which puts the spacecraft from the parking orbit into a hyperbolic escape trajectory. Then another impulsive change is required to enter the transfer orbit about the Sun (Hohmann transfer). Mars approach can be achieved in different ways, always via impulsive decelerations. In this case, the initial burn transfers the spacecraft from heliocentric orbit, into an elongated elliptical orbit about Mars. Successive decelerations bring the Surveyor to a circular orbit. In this course, we will concentrate on the midcourse and terminal phases of guidance.... Guidance Classification In addition to phases of travel, guidance methods can be classified according to the type of information, the amount of closed loop interaction, and methods for actually deriving the guidance law itself. We can start from the guidance algorithms described in the first flowchart. In a preset guidance algorithm, all information is stored on the vehicle at launch. In particular, 7
8 target information is not updated, although the guidance system is capable of maintaining the course and reducing course error with the help of the autopilot. Direct guidance methods are characterized by the fact that information about the target is received at launch, as well as during the flight. This feature allows interception of moving targets and the implementation makes the difference between one guidance law and another. Direct guidance methods are well established, and further improvement will depend only on hardware development. Command guidance is the guidance of a vehicle by means of electronic commands generated outside. The basic concept is to study the relative position and velocity of vehicletarget, and then uplink the information so that intercept at some position is achieved. The information can be given to the vehicle in various ways (radar station, radio frequency, laser beam, uncoiled wire, etc.). The main advantage of this solution is that there is no need of onboard seeker; the main limitation is the influence of noise on the data transmission to the vehicle. CLOS is a derivative of command guidance, where the lineofsight information is in some way given to the vehicle, thus improving the capability of intercept. The vehicle approaches the target along a line joining a control point and the target itself. At the control point, the target is tracked by a radar (or other method), while the vehicle is tracked by the same radar or by some other point. If the vehicle can maintain the straight line between tracker and target (LOS), it will intercept. The input to the guidance algorithm is therefore the relative error vector and its rate, producing an appropriate compensating acceleration. This can be obtained by applied different control methods, from classical control, to optimal control, from feedback linearization, to neural and fuzzy systems. Beamrider guidance is a threepoint guidance where the guidance algorithm maintains the vehicle inside a beam directed to the target. The beam is usually either radar or laser generated. Homing guidance is typically applied to the terminal phase of the guidance trajectory. In this case, the vehicle is guided by information coming directly from the target, in three 8
9 main solutions: active, semiactive, and passive. In active homing, the vehicle has a transmitter and a receiver and illuminates the target, thus obtaining information about its position and velocity. In semiactive homing, the illumination comes from a source outside the vehicle. In passive homing, the target itself is known due to some special radiating information that is unique to the target itself (heat, electromagnetic waves, etc.). In all three cases, however, the vehicle must have a seeker (system that is capable of performing the homing information measurement and processing it to the guidance law. In order to intercept a target or to reach a specified point precisely, the vehicle must constantly travel in the appropriate direction, which is achieved by a guidance law built into the global guidance processor. The synthesis of a guidance law is obtained using different control methods, and to be complete it needs to be combined with appropriate filtering and estimation, which reduce the influence of noise sources. A schematic of currently used guidance laws is shown below. Although similar in principle (LOS information is crucial to the success of the guidance), there are differences with respect to a particular aspect of intercept: the capability of achieving intercept in the presence of a maneuverable target and/or a target having speed higher than that of the vehicle. For a brief explanation of the differences, we refer to the figure below, which illustrates the variables of interest. As we said before, the main purpose of the guidance law is to provide an appropriate commanded acceleration history, which will guarantee intercept. Based on the engagement geometry, we can compute the vehicle acceleration as: 9
10 R σ&& A = σ & V + + V& tanσ m m m m lead cos σlead After some manipulations (the details will be given later), the kinematic transfer function becomes θm(s) 1 1 = A m(s) s ( Vm V& mr m / R& m) + ( VmR m / R& m) s Rms The LOS loop becomes unstable under small perturbations, and requires a stabilizing lead compensator as shown in the block diagram: The homing guidance system, which contrasts with LOS guidance, is designated as a twopoint guidance system and it is implemented mostly as LOS rate guidance. The next block diagram shows a block diagram implementation of the latter solution. Since there is only a pole at the origin, instability issues are less critical, at least until near intercept. Since however it is desirable to keep the guidance loop bandwidth as small as possible, a solution called Proportional Navigation is usually adopted. The 10
11 implementation of LOS rate guidance in terms of commanded flight path angle is given by: γ & = λσ& Where λ is the socalled navigation gain. c And the proportional navigation implementation loop is shown here: Pursuit guidance was the first implemented time wise. It resembles the classical houndhare course. The vehicle aims directly at the target during the entire engagement. While it permits a simple implementation and is less sensitive to noise, pursuit guidance is somewhat impractical against highly moving targets, resulting in energy consuming tail chase. This is NOT the case for instance for ships and underwater vehicles, where the speed advantage is much greater. A less demanding course is the one that predicts the target path and aims the vehicle to the predicted intercept point. This works well if the predictor scheme is accurate, and 11
12 the autopilot is commanded to keep the direction (LOS) to the target constant, resulting in the socalled constant bearing guidance shown below. Proportional Navigation Guidance is a derivation of constant bearing guidance, and will be studied in detail in the next section. The basic concept is to make the vehicle s heading rate proportional to the LOS rate from vehicle to target, yielding a trajectory like in the figure. The term Advanced guidance identifies all new developments in the field of homing guidance. Here the problem is generalized to include prediction, and optimality is taken into account in many possible ways (calculus of variations optimization, constrained nonlinear optimization and onoff control, differential games, fuzzy logic, robust multivariable control methods, etc.)..3. Proportional Navigation Guidance (PNG) Because of the widespread use, and connection to the optimal solution of a two point boundary value problem in certain cases, PNG is examined in detail in this section Development The base principle common to all types of PNG is that of controlling the direction of the vehicle s velocity vector, with a maneuver speed proportional to the LOS rate. It appears that the principle was known and used by pirates in order to achieve collision course with merchant ships starting from More recently, the method was studied by Germany at the end of WW II, and by the USA in the same period. Due to its physical simplicity, the method was implemented in the early 50, and first published in 1948 in the Journal of Applied Physics. Mathematical derivation of its optimality under certain conditions followed only 0 years later with the work by Arthur Bryson. PNG gives the possibility of defining the acceleration command to give the vehicle for intercept. While the magnitude of the acceleration is common to most of the variants and proportional to LOS rate, the commanded acceleration direction changes according to different solutions. In addition, augmented PNG has a term that takes into account target motion capabilities. 1
13 Pure PNG has the acceleration command normal to the velocity of the vehicle; True PNG has the direction normal to the LOS, In modified True PNG, the acceleration direction is still normal to the LOS, but the magnitude is proportional to the product between LOS rate and closing speed (relative speed), Ideal PNG has the direction of the acceleration normal to the closing speed, Generalized True PNG has the direction of the acceleration at a specified angle with respect to the LOS..3.. Optimality The optimality of PNG was proved in 1969 by Bryson, by casting the scenario as a differential game between two players: pursuer and evader. Differential games are a particular set of optimization problems involving players with conflicting interest. As imagined, they involve continuous differential equations and functional performance criteria. Game theory and optimization are very close in mathematical terms, but they were established independently. In guidance problems requiring intercept, the two players are the vehicle and target (pursuer and evader), and their conflicting interest is the solution of a MinMax optimization problem. In mathematical form we have the following problem: Given the dynamic system x& = f(x,u,v,t) x(t ) = x 0 0 (1) Terminal constraints And the performance criterion ( x(t ),t ) Ψ =0 () f f 0 0 Find the couple ( u,v ) t f ( ) [ ] J x(t ),t L(x,u,v,t) dt =Φ f f + t0 such that (3) 13
14 ( 0 ) ( 0 0 ) ( 0 ) J u,v J u,v J u,v (4) The first order necessary conditions are the same as calculus of variations; therefore we have for this minmax problem: =λ + T H f L λ & T = T H, λ x (t) =Φ f x(t f ) H H u v = 0 = 0 (5) Or H 0 = maxminh v u (6) In order to obtain sufficiency of the solution, we also require the existence of a gametheoretic saddle point, which not necessarily exists unless H is separable. Fortunately in our applications this is not the case since both f and L are in general separable. Note that for the function L(u,v ) we have: Game theoretic saddle point Lu = 0 Lv = 0 L L uu Calculus saddle point L = 0 L = 0 u 0 vv 0 LuuLvv ( Luv ) Another interesting point is the evaluation of optimization (4) as an open loop or closed loop strategy. In fact from (4) it is not clear if one player strategy is optimal based on the other s operating in an open loop or closed loop fashion. Although this is not part of the course, it suffices to notice that optimality of one player s behavior is much more stringent when the other operates in feedback fashion. Consider now the standard intercept planar problem based on the next figure, reformulated as a differential game with limits on the vehicles acceleration. 0 v 14
15 In a pursuitevasion game, the pursuer s control is its acceleration a(t) p normal to the initial LOS, and the evader does the same with its acceleration a(t). t In the following, vector and scalar quantities will be understood from the context. If v(t) and y(t) are the relative velocity and displacement normal to the initial LOS, the equations of motion are: v& = ap at v(t ) = v y& = v y(t f ) = (7) The pursuer wishes to minimize the terminal miss y(t f ), while the evader tends to maximize it, therefore a feasible performance index is: 1 J = y(t ) [ ] f (8) The controls are bounded as: at ap a pm,a pm > a tm a tm (9) The Hamiltonian for the problem is: H = λv(ap a t ) +λ yv. The adjoint equations and optimality conditions are readily obtained as: λ & v = λy λ v(t f ) = 0 λ & y λ y(t f ) = y(t f ) a(t) = a sgnλ a(t) t = atmsgnλv p pm v Solving the EulerLagrange equations, we can determine an expression for y(t f ) as 1 v 0 ( tf t0 ) ( apm atm) tf t0 y(t f ) = 1 v ( tf 0 t0 ) ( apm atm) tf t0 However no solution exists for, v 0 if > 1 ( tf t0 )( apm atm) v 0 if < 1 ( tf t0 )( apm atm) v 0 1< if < 1 ( tf t0 )( apm atm) In fact, for this range of initial conditions, the pursuer can always achieve zero miss distance, for instance choosing its acceleration such that: 15
16 v(t ) a p(t) = a(t) t + t t f. The term t f t is also called time to go or t go. Let us consider now a linear quadratic differential game, with a quadratic performance index that contains the acceleration limitations, in a 3D environment. The equations of motion are: v& p = fp + ap v& t = ft + at, & rp = vp r & t = vt (10) Assuming the gravity contribution f the same, the only variable of interest is the relative position vector r(t) r(t). Consider the performance index: p t t { p t p t } b J = r (t) r (t) r (t) r (t) + 1 f 1 1 { p ( p p) t ( t t) } 0 c a a c a a dt (11) Using (5) yields: c (t t) r (t) r (t) + (v (t) v (t))(t t) p f p t p t f ap = 3 1 (tf t) + (cp c t ) b 3 (1) ct at = ap cp In (1) we note that: If cp > ct the feedback control gain has always the same sign, If cp < ct the feedback control gain changes sign when and t f sufficiently large. 1 b (t t) 3 3 f + (cp c t ) = 0 (13) Equation (13) is actually the conjugate point condition for the optimization problem (10) (11), that is the relationship for which sufficient conditions for global minimum are not satisfied according to the second variation properties. Therefore case cp < ct is not longer optimal, which means that intercept is not possible when cp < ct and evident for b=. At this point we can select the optimal solution corresponding to cp > ct. With b=, the pursuer s control strategy becomes: 16
17 3 a = r (t) r(t) (v (t) v (t))(t t) c + (t t) p p t p t f p 1 f c t (14) Now let the pursuer and the follower be on a nominal collision course with range and closing velocity respectively given by R Vc R = t t f Let the lateral deviation from collision be given by r(t) p r(t) t as shown below From the engagement geometry, the lateral acceleration to be applied by the pursuer from (14) is given by: a LAT 3 = Vcσ& (15) c t 1 cp Which is simply Proportional Navigation with an effective navigation gain of 3 N = ct 1 cp Based on experimental results, the best value of N has been found to be between 3 and 5. Note that N=3 corresponds to non maneuverable evader (c t =0), whereas N=5 corresponds to a weighting ratio c t /c p =/5. 17
18 .3.3. DScenario In this section, we will limit ourselves to a twodimensional scenario, which is the most common in guidance problems. Nomenclature: A,A m A,A m t t Vehicle and target accelerations Vehicle and target velocities γm, γ t Vehicle and target flight path angles σ LOS angle R Range Vc = R & Closing velocity (relative velocity along the inst. LOS) The inertial reference system has the abscissa in the down range direction and the ordinate either altitude (depth) or cross range. The angle α = L+ He = γm σ has two components: L is the lead angle that is the theoretical value for a collision course (no additional acceleration commands are required for intercept). H e is known as heading error, or the initial deviation from the collision triangle. The equations of motion of the geometry above can be written in Cartesian form: x&& = A sinγ z&& m = Amcosγm x&& t = At sinγt z&& t = At cosγt m m m γ m =tan z& x & z& 1 m m γ t=tan R= ( x x ) +( z  z ) 1 zo zm σ=tan x o x m 1 t x & t (16) o m o m 18
19 ( xt x m)( x t x m)+(ztz m )(z o  z m) V= c ˆ  R & & & & & =  R ( xt x m)(z & o  z & m)( ztz m )( x & t x & m) σ & =  R (17) Or in Polar form: R& = Vc = Vt cos( γt σ) Vmcos( γm σ) Vt sin( γt σ) Vmsin( γm σ) σ & = R R && = R σ & + Amsin( γm σ) At sin( γt σ) σ & R& + At cos( γt σ) Amcos( γm σ ) σ && = R Am γ & m = V m A γ o & t = V t (18) (19) In the previous section, we have seen the mathematical proof of optimality from the standpoint of a differential game approach. Consider now the collision triangle ( R & < 0 ) below: If the pursuer velocity is constant, the evader is not maneuvering, and the LOS does not rotate in the inertial space, collision will occur. Consider in fact the second of (18), differentiating and substituting back into the first yields: Rσ+ && R NVmcos( m) & + σ γ = V& sin( σ γ ) +γ& Vcos( σ γ ) Vsin( & σ γ ) m m t t t t t (0) From (15), for non maneuvering target, γ & m = Nσ& and pursuer constant speed, the RHS of (0) becomes equal to zero and (0) is a homogeneous differential equation in LOS rate. 19
20 Set the navigation ratio N as: Λ R& = Λ Λ > n N n, Vmcos( σ γm) n (1) And substituted in (0), yields: [ ] Rσ&& R& Λ σ & = 0 () n This has an analytical solution R σ=σ & & 0 R0 Λn Λn R γ & m = γ & m0 R0 (3) The solution tends to zero (intercept) for R & < 0 Λ n =3, and asymptotic decrease for Λn.. Linear decrease is achieved for NOTE: From the PNG, it is possible to go back to the expressions for pursuit and constant bearing guidance laws. Recall that in pursuit guidance, the vehicle aims directly at the target during the entire engagement. Assuming the usual constraints on pursuer and evader speed ( Vt = const. γ t = const. = 0 ;Vm = const. ), the guidance law assumes the form: γ m = σ (4) Am = Vmσ& From (18) R& = Vt cos γt Vm (5) R σ & = Vt sin γ t But σ& =0 if σ=0 or σ=π (headon, or tailon), which means that the pursuer must always be turning. The case of constant bearing guidance is even closer to PNG. It derives directly from the intercept triangle conditions, and it requires: Example σ& 0 (6) Consider the following engagement; the pursuer has a heading angle of 0 degrees. 0
21 The results in terms of PNG guidance are shown next Twodimensional tactical missiletarget engagement simulation 15 Twodimensional tactical missiletarget engagement simulation red > N=4 green> N=5 Altitude (Ft) Acceleration of missle (G) Downrange (Ft) x Time (sec) Consider the second engagement, with the evader having a sudden acceleration (maneuverable target) The next two figures show the vehicles trajectories, as well as pursuer acceleration profiles for the two navigation ratios of 4 and 5. Even if the optimality of PNG is not guaranteed for maneuvering targets, in this example intercept occurs. 1.5 x 104 Twodimensional tactical missiletarget engagement simulation 4.5 Twodimensional tactical missiletarget engagement simulation Altitude (Ft) Red> N=4 Acceleration of missle (G) Downrange (Ft) x Time (sec).3.4. Linear Approximation 1
22 As usual, many important feature of dynamic systems behaviour can be inferred by a linear analysis. In the previous sections we introduced PNG and evaluated its optimality, given some engagement assumptions. In simulation, however, the intercept was successful even in the case of maneuverable target (at least for the case shown). A linear view of the problem will help understanding why. Recall the D scenario described in section.3.3.: Based on the above geometry, consider the relative acceleration normal to the reference axis 1, A = y&& = A cosβ A cos σ R R t m Using small angle approximation (small flight path angles) we have: AR At Am yr σ R Treating the closing velocity as a positive constant, since R 0 at and of the flight, we can use the linear expression for the range: R = V c(tf t ) = Vct go,where tf = const.
23 The linearized miss distance, or relative separation ad the end of the engagement, is y (t ) R f If we consider the previous two examples, we have: From the time histories we can see that in the case of non maneuvering target the linear approximation is excellent, while an overestimate is present when the target is maneuvering because linearized guidance assumes the target maintaining the same acceleration vector all throughout the flight. Since the problem is linearized, a closed form solution can be obtained (for the two examples above). Recall that Then: HE yr = miss dis tan ce initial heading error At Am = y && R, MAN yr = miss dis tan ce t arg et maneuverable y&& = NV σ& y&& NV A HE R c MAN R = cσ+ & t Solving for the transition matrix and using the initial conditions, yields: N HE HE Vm N t Amc = 1 tf tf A N t = 1 1 N MAN mc N t f A t (*) An important aspect of the linearized approach is the physically insight gained by examining the solution, and the definition of Zero Effort Miss (ZEM), a parameter important also in other types of guidance laws. Definition: Define ZEM as the distance the pursuer would miss the evader, if the evader continued along its present course, and the pursuer made no further corrective maneuvers. 3
24 From the above equations, and the standard form of PNG (15) we have: Amc = NVc σ& R = V (t t) = V t yr σ R c f c go (7) Substituting and differentiating yields: From which: d y =σ & R Amc NVc dt V c (t f t ) N Amc = yr + y& Rt go t go,zem = yr + y& Rtgo Amc = NVc σ& (8) In (8) we can see that the term in brackets is nothing but the ZEM, therefore PNG, when linearization applies, can be thought as a guidance law which is also proportional to the ZEM and inversely proportional to the square of the timetogo. The Adjoint Method The adjoint method is a powerful optimization technique dating to Volterra (1870), and used originally for artillery studies. It combines gradient based optimization with a linearized guidance loop dynamics. The above diagram, based on the preceding discussions, is the simplest form of a linearized homing loop. We can see that noise processing and autopilot are considered perfect, and the LOS rate information is computed perfectly by the seeker; this absence 4
Optimal Design of αβ(γ) Filters
Optimal Design of (γ) Filters Dirk Tenne Tarunraj Singh, Center for Multisource Information Fusion State University of New York at Buffalo Buffalo, NY 426 Abstract Optimal sets of the smoothing parameter
More informationOrbital Mechanics. Angular Momentum
Orbital Mechanics The objects that orbit earth have only a few forces acting on them, the largest being the gravitational pull from the earth. The trajectories that satellites or rockets follow are largely
More informationLecture L6  Intrinsic Coordinates
S. Widnall, J. Peraire 16.07 Dynamics Fall 2009 Version 2.0 Lecture L6  Intrinsic Coordinates In lecture L4, we introduced the position, velocity and acceleration vectors and referred them to a fixed
More informationRobotics. Lecture 3: Sensors. See course website http://www.doc.ic.ac.uk/~ajd/robotics/ for up to date information.
Robotics Lecture 3: Sensors See course website http://www.doc.ic.ac.uk/~ajd/robotics/ for up to date information. Andrew Davison Department of Computing Imperial College London Review: Locomotion Practical
More informationSpacecraft Dynamics and Control. An Introduction
Brochure More information from http://www.researchandmarkets.com/reports/2328050/ Spacecraft Dynamics and Control. An Introduction Description: Provides the basics of spacecraft orbital dynamics plus attitude
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationEnhancing the SNR of the Fiber Optic Rotation Sensor using the LMS Algorithm
1 Enhancing the SNR of the Fiber Optic Rotation Sensor using the LMS Algorithm Hani Mehrpouyan, Student Member, IEEE, Department of Electrical and Computer Engineering Queen s University, Kingston, Ontario,
More informationG U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M
G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M CONTENTS Foreword... 2 Forces... 3 Circular Orbits... 8 Energy... 10 Angular Momentum... 13 FOREWORD
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationForce/position control of a robotic system for transcranial magnetic stimulation
Force/position control of a robotic system for transcranial magnetic stimulation W.N. Wan Zakaria School of Mechanical and System Engineering Newcastle University Abstract To develop a force control scheme
More informationLecture L17  Orbit Transfers and Interplanetary Trajectories
S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L17  Orbit Transfers and Interplanetary Trajectories In this lecture, we will consider how to transfer from one orbit, to another or to
More informationRobot Perception Continued
Robot Perception Continued 1 Visual Perception Visual Odometry Reconstruction Recognition CS 685 11 Range Sensing strategies Active range sensors Ultrasound Laser range sensor Slides adopted from Siegwart
More informationAstromechanics TwoBody Problem (Cont)
5. Orbit Characteristics Astromechanics TwoBody Problem (Cont) We have shown that the in the twobody problem, the orbit of the satellite about the primary (or viceversa) is a conic section, with the
More informationOrigins of the Unusual Space Shuttle Quaternion Definition
47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition 58 January 2009, Orlando, Florida AIAA 200943 Origins of the Unusual Space Shuttle Quaternion Definition
More informationPower Electronics. Prof. K. Gopakumar. Centre for Electronics Design and Technology. Indian Institute of Science, Bangalore.
Power Electronics Prof. K. Gopakumar Centre for Electronics Design and Technology Indian Institute of Science, Bangalore Lecture  1 Electric Drive Today, we will start with the topic on industrial drive
More informationExperiment #1, Analyze Data using Excel, Calculator and Graphs.
Physics 182  Fall 2014  Experiment #1 1 Experiment #1, Analyze Data using Excel, Calculator and Graphs. 1 Purpose (5 Points, Including Title. Points apply to your lab report.) Before we start measuring
More informationAlgebra 1 2008. Academic Content Standards Grade Eight and Grade Nine Ohio. Grade Eight. Number, Number Sense and Operations Standard
Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express
More informationAstrodynamics (AERO0024)
Astrodynamics (AERO0024) 6. Interplanetary Trajectories Gaëtan Kerschen Space Structures & Systems Lab (S3L) Course Outline THEMATIC UNIT 1: ORBITAL DYNAMICS Lecture 02: The TwoBody Problem Lecture 03:
More informationPID Control. Chapter 10
Chapter PID Control Based on a survey of over eleven thousand controllers in the refining, chemicals and pulp and paper industries, 97% of regulatory controllers utilize PID feedback. Desborough Honeywell,
More informationMOBILE ROBOT TRACKING OF PREPLANNED PATHS. Department of Computer Science, York University, Heslington, York, Y010 5DD, UK (email:nep@cs.york.ac.
MOBILE ROBOT TRACKING OF PREPLANNED PATHS N. E. Pears Department of Computer Science, York University, Heslington, York, Y010 5DD, UK (email:nep@cs.york.ac.uk) 1 Abstract A method of mobile robot steering
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationIntroduction to Engineering System Dynamics
CHAPTER 0 Introduction to Engineering System Dynamics 0.1 INTRODUCTION The objective of an engineering analysis of a dynamic system is prediction of its behaviour or performance. Real dynamic systems are
More informationAP1 Oscillations. 1. Which of the following statements about a springblock oscillator in simple harmonic motion about its equilibrium point is false?
1. Which of the following statements about a springblock oscillator in simple harmonic motion about its equilibrium point is false? (A) The displacement is directly related to the acceleration. (B) The
More informationEDUMECH Mechatronic Instructional Systems. Ball on Beam System
EDUMECH Mechatronic Instructional Systems Ball on Beam System Product of Shandor Motion Systems Written by Robert Hirsch Ph.D. 9989 All Rights Reserved. 999 Shandor Motion Systems, Ball on Beam Instructional
More informationSection 4: The Basics of Satellite Orbits
Section 4: The Basics of Satellite Orbits MOTION IN SPACE VS. MOTION IN THE ATMOSPHERE The motion of objects in the atmosphere differs in three important ways from the motion of objects in space. First,
More informationAdvantages of Autotuning for Servomotors
Advantages of for Servomotors Executive summary The same way that 2 years ago computer science introduced plug and play, where devices would selfadjust to existing system hardware, industrial motion control
More informationUSING MS EXCEL FOR DATA ANALYSIS AND SIMULATION
USING MS EXCEL FOR DATA ANALYSIS AND SIMULATION Ian Cooper School of Physics The University of Sydney i.cooper@physics.usyd.edu.au Introduction The numerical calculations performed by scientists and engineers
More informationRobotics and Automation Blueprint
Robotics and Automation Blueprint This Blueprint contains the subject matter content of this Skill Connect Assessment. This Blueprint does NOT contain the information one would need to fully prepare for
More informationIn mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.
MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target
More information3.2 Sources, Sinks, Saddles, and Spirals
3.2. Sources, Sinks, Saddles, and Spirals 6 3.2 Sources, Sinks, Saddles, and Spirals The pictures in this section show solutions to Ay 00 C By 0 C Cy D 0. These are linear equations with constant coefficients
More informationTCOM 370 NOTES 994 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS
TCOM 370 NOTES 994 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS 1. Bandwidth: The bandwidth of a communication link, or in general any system, was loosely defined as the width of
More informationLecture L2  Degrees of Freedom and Constraints, Rectilinear Motion
S. Widnall 6.07 Dynamics Fall 009 Version.0 Lecture L  Degrees of Freedom and Constraints, Rectilinear Motion Degrees of Freedom Degrees of freedom refers to the number of independent spatial coordinates
More informationFURTHER VECTORS (MEI)
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level  MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: 97 Mathematics
More informationE190Q Lecture 5 Autonomous Robot Navigation
E190Q Lecture 5 Autonomous Robot Navigation Instructor: Chris Clark Semester: Spring 2014 1 Figures courtesy of Siegwart & Nourbakhsh Control Structures Planning Based Control Prior Knowledge Operator
More informationPath Tracking for a Miniature Robot
Path Tracking for a Miniature Robot By Martin Lundgren Excerpt from Master s thesis 003 Supervisor: Thomas Hellström Department of Computing Science Umeå University Sweden 1 Path Tracking Path tracking
More informationIMU Components An IMU is typically composed of the following components:
APN064 IMU Errors and Their Effects Rev A Introduction An Inertial Navigation System (INS) uses the output from an Inertial Measurement Unit (IMU), and combines the information on acceleration and rotation
More informationElectronics for Analog Signal Processing  II Prof. K. Radhakrishna Rao Department of Electrical Engineering Indian Institute of Technology Madras
Electronics for Analog Signal Processing  II Prof. K. Radhakrishna Rao Department of Electrical Engineering Indian Institute of Technology Madras Lecture  18 Wideband (Video) Amplifiers In the last class,
More informationLecture 8 : Dynamic Stability
Lecture 8 : Dynamic Stability Or what happens to small disturbances about a trim condition 1.0 : Dynamic Stability Static stability refers to the tendency of the aircraft to counter a disturbance. Dynamic
More informationOrbital Dynamics with Maple (sll  v1.0, February 2012)
Orbital Dynamics with Maple (sll  v1.0, February 2012) Kepler s Laws of Orbital Motion Orbital theory is one of the great triumphs mathematical astronomy. The first understanding of orbits was published
More informationDirichlet forms methods for error calculus and sensitivity analysis
Dirichlet forms methods for error calculus and sensitivity analysis Nicolas BOULEAU, Osaka university, november 2004 These lectures propose tools for studying sensitivity of models to scalar or functional
More informationOrbits of the LennardJones Potential
Orbits of the LennardJones Potential Prashanth S. Venkataram July 28, 2012 1 Introduction The LennardJones potential describes weak interactions between neutral atoms and molecules. Unlike the potentials
More informationMATH. ALGEBRA I HONORS 9 th Grade 12003200 ALGEBRA I HONORS
* Students who scored a Level 3 or above on the Florida Assessment Test Math Florida Standards (FSAMAFS) are strongly encouraged to make Advanced Placement and/or dual enrollment courses their first choices
More informationEðlisfræði 2, vor 2007
[ Assignment View ] [ Pri Eðlisfræði 2, vor 2007 28. Sources of Magnetic Field Assignment is due at 2:00am on Wednesday, March 7, 2007 Credit for problems submitted late will decrease to 0% after the deadline
More informationActive Vibration Isolation of an Unbalanced Machine Spindle
UCRLCONF206108 Active Vibration Isolation of an Unbalanced Machine Spindle D. J. Hopkins, P. Geraghty August 18, 2004 American Society of Precision Engineering Annual Conference Orlando, FL, United States
More informationPositive Feedback and Oscillators
Physics 3330 Experiment #6 Fall 1999 Positive Feedback and Oscillators Purpose In this experiment we will study how spontaneous oscillations may be caused by positive feedback. You will construct an active
More informationWe can display an object on a monitor screen in three different computermodel forms: Wireframe model Surface Model Solid model
CHAPTER 4 CURVES 4.1 Introduction In order to understand the significance of curves, we should look into the types of model representations that are used in geometric modeling. Curves play a very significant
More informationUnderstanding Poles and Zeros
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function
More informationTime Domain and Frequency Domain Techniques For Multi Shaker Time Waveform Replication
Time Domain and Frequency Domain Techniques For Multi Shaker Time Waveform Replication Thomas Reilly Data Physics Corporation 1741 Technology Drive, Suite 260 San Jose, CA 95110 (408) 2168440 This paper
More informationFigure 1. The Ball and Beam System.
BALL AND BEAM : Basics Peter Wellstead: control systems principles.co.uk ABSTRACT: This is one of a series of white papers on systems modelling, analysis and control, prepared by Control Systems Principles.co.uk
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationCONTRIBUTIONS TO THE AUTOMATIC CONTROL OF AERIAL VEHICLES
1 / 23 CONTRIBUTIONS TO THE AUTOMATIC CONTROL OF AERIAL VEHICLES MINH DUC HUA 1 1 INRIA Sophia Antipolis, AROBAS team I3SCNRS Sophia Antipolis, CONDOR team Project ANR SCUAV Supervisors: Pascal MORIN,
More informationParameter identification of a linear single track vehicle model
Parameter identification of a linear single track vehicle model Edouard Davin D&C 2011.004 Traineeship report Coach: dr. Ir. I.J.M. Besselink Supervisors: prof. dr. H. Nijmeijer Eindhoven University of
More informationCHAPTER 1 INTRODUCTION
CHAPTER 1 INTRODUCTION 1.1 Background of the Research Agile and precise maneuverability of helicopters makes them useful for many critical tasks ranging from rescue and law enforcement task to inspection
More informationCHAPTER 5 PREDICTIVE MODELING STUDIES TO DETERMINE THE CONVEYING VELOCITY OF PARTS ON VIBRATORY FEEDER
93 CHAPTER 5 PREDICTIVE MODELING STUDIES TO DETERMINE THE CONVEYING VELOCITY OF PARTS ON VIBRATORY FEEDER 5.1 INTRODUCTION The development of an active trap based feeder for handling brakeliners was discussed
More informationPhysics Kinematics Model
Physics Kinematics Model I. Overview Active Physics introduces the concept of average velocity and average acceleration. This unit supplements Active Physics by addressing the concept of instantaneous
More informationPrecise Modelling of a Gantry Crane System Including Friction, 3D Angular Swing and Hoisting Cable Flexibility
Precise Modelling of a Gantry Crane System Including Friction, 3D Angular Swing and Hoisting Cable Flexibility Renuka V. S. & Abraham T Mathew Electrical Engineering Department, NIT Calicut Email : renuka_mee@nitc.ac.in,
More informationCommon Core Unit Summary Grades 6 to 8
Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity 8G18G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationOPRE 6201 : 2. Simplex Method
OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2
More informationDevelopment and optimization of a hybrid passive/active liner for flow duct applications
Development and optimization of a hybrid passive/active liner for flow duct applications 1 INTRODUCTION Design of an acoustic liner effective throughout the entire frequency range inherent in aeronautic
More informationDEOS. Deutsche Orbitale Servicing Mission. The Inflight Technology Demonstration of Germany s Robotics Approach to Service Satellites
DEOS Deutsche Orbitale Servicing Mission The Inflight Technology Demonstration of Germany s Robotics Approach to Service Satellites B. Sommer, K. Landzettel, T. Wolf, D. Reintsema, German Aerospace Center
More informationDRAFT. Further mathematics. GCE AS and A level subject content
Further mathematics GCE AS and A level subject content July 2014 s Introduction Purpose Aims and objectives Subject content Structure Background knowledge Overarching themes Use of technology Detailed
More informationMechanics lecture 7 Moment of a force, torque, equilibrium of a body
G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and
More informationAttitude Control and Dynamics of Solar Sails
Attitude Control and Dynamics of Solar Sails Benjamin L. Diedrich A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics & Astronautics University
More information2After completing this chapter you should be able to
After completing this chapter you should be able to solve problems involving motion in a straight line with constant acceleration model an object moving vertically under gravity understand distance time
More informationOnboard electronics of UAVs
AARMS Vol. 5, No. 2 (2006) 237 243 TECHNOLOGY Onboard electronics of UAVs ANTAL TURÓCZI, IMRE MAKKAY Department of Electronic Warfare, Miklós Zrínyi National Defence University, Budapest, Hungary Recent
More informationEE 402 RECITATION #13 REPORT
MIDDLE EAST TECHNICAL UNIVERSITY EE 402 RECITATION #13 REPORT LEADLAG COMPENSATOR DESIGN F. Kağan İPEK Utku KIRAN Ç. Berkan Şahin 5/16/2013 Contents INTRODUCTION... 3 MODELLING... 3 OBTAINING PTF of OPEN
More informationA PHOTOGRAMMETRIC APPRAOCH FOR AUTOMATIC TRAFFIC ASSESSMENT USING CONVENTIONAL CCTV CAMERA
A PHOTOGRAMMETRIC APPRAOCH FOR AUTOMATIC TRAFFIC ASSESSMENT USING CONVENTIONAL CCTV CAMERA N. Zarrinpanjeh a, F. Dadrassjavan b, H. Fattahi c * a Islamic Azad University of Qazvin  nzarrin@qiau.ac.ir
More informationPerformance. 13. Climbing Flight
Performance 13. Climbing Flight In order to increase altitude, we must add energy to the aircraft. We can do this by increasing the thrust or power available. If we do that, one of three things can happen:
More informationChapter 11 Equilibrium
11.1 The First Condition of Equilibrium The first condition of equilibrium deals with the forces that cause possible translations of a body. The simplest way to define the translational equilibrium of
More informationMoral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania
Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 PrincipalAgent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically
More informationThe Kinetic Theory of Gases Sections Covered in the Text: Chapter 18
The Kinetic Theory of Gases Sections Covered in the Text: Chapter 18 In Note 15 we reviewed macroscopic properties of matter, in particular, temperature and pressure. Here we see how the temperature and
More information6 J  vector electric current density (A/m2 )
Determination of Antenna Radiation Fields Using Potential Functions Sources of Antenna Radiation Fields 6 J  vector electric current density (A/m2 ) M  vector magnetic current density (V/m 2 ) Some problems
More informationLecture L5  Other Coordinate Systems
S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L5  Other Coordinate Systems In this lecture, we will look at some other common systems of coordinates. We will present polar coordinates
More informationBasic Principles of Inertial Navigation. Seminar on inertial navigation systems Tampere University of Technology
Basic Principles of Inertial Navigation Seminar on inertial navigation systems Tampere University of Technology 1 The five basic forms of navigation Pilotage, which essentially relies on recognizing landmarks
More informationState Newton's second law of motion for a particle, defining carefully each term used.
5 Question 1. [Marks 28] An unmarked police car P is, travelling at the legal speed limit, v P, on a straight section of highway. At time t = 0, the police car is overtaken by a car C, which is speeding
More informationHello, my name is Olga Michasova and I present the work The generalized model of economic growth with human capital accumulation.
Hello, my name is Olga Michasova and I present the work The generalized model of economic growth with human capital accumulation. 1 Without any doubts human capital is a key factor of economic growth because
More informationOn Motion of Robot EndEffector using the Curvature Theory of Timelike Ruled Surfaces with Timelike Directrix
Malaysian Journal of Mathematical Sciences 8(2): 89204 (204) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: http://einspem.upm.edu.my/journal On Motion of Robot EndEffector using the Curvature
More informationRotation: Moment of Inertia and Torque
Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn
More informationOPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 345 354 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN E.V. Grigorieva Department of Mathematics
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationAlignment Laser System.
 O T 6 0 0 0 Alignment Laser System. The OT6000. Multi Target,Two Dimensional Alignment. Introducing the most powerful way to measure alignment at distances up to 300 feet. The OT6000 Alignment Laser
More informationKINEMATICS OF PARTICLES RELATIVE MOTION WITH RESPECT TO TRANSLATING AXES
KINEMTICS OF PRTICLES RELTIVE MOTION WITH RESPECT TO TRNSLTING XES In the previous articles, we have described particle motion using coordinates with respect to fixed reference axes. The displacements,
More information19 LINEAR QUADRATIC REGULATOR
19 LINEAR QUADRATIC REGULATOR 19.1 Introduction The simple form of loopshaping in scalar systems does not extend directly to multivariable (MIMO) plants, which are characterized by transfer matrices instead
More informationRS platforms. Fabio Dell Acqua  Gruppo di Telerilevamento
RS platforms Platform vs. instrument Sensor Platform Instrument The remote sensor can be ideally represented as an instrument carried by a platform Platforms Remote Sensing: Groundbased airborne spaceborne
More informationCopyright 2011 Casa Software Ltd. www.casaxps.com
Table of Contents Variable Forces and Differential Equations... 2 Differential Equations... 3 Second Order Linear Differential Equations with Constant Coefficients... 6 Reduction of Differential Equations
More informationLecture 2 Linear functions and examples
EE263 Autumn 200708 Stephen Boyd Lecture 2 Linear functions and examples linear equations and functions engineering examples interpretations 2 1 Linear equations consider system of linear equations y
More informationDesignSimulationOptimization Package for a Generic 6DOF Manipulator with a Spherical Wrist
DesignSimulationOptimization Package for a Generic 6DOF Manipulator with a Spherical Wrist MHER GRIGORIAN, TAREK SOBH Department of Computer Science and Engineering, U. of Bridgeport, USA ABSTRACT Robot
More informationPenn State University Physics 211 ORBITAL MECHANICS 1
ORBITAL MECHANICS 1 PURPOSE The purpose of this laboratory project is to calculate, verify and then simulate various satellite orbit scenarios for an artificial satellite orbiting the earth. First, there
More informationEE4367 Telecom. Switching & Transmission. Prof. Murat Torlak
Path Loss Radio Wave Propagation The wireless radio channel puts fundamental limitations to the performance of wireless communications systems Radio channels are extremely random, and are not easily analyzed
More informationSOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS  VELOCITY AND ACCELERATION DIAGRAMS
SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS  VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering
More informationComputer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science  Technion. An Example.
An Example 2 3 4 Outline Objective: Develop methods and algorithms to mathematically model shape of real world objects Categories: WireFrame Representation Object is represented as as a set of points
More information8.2 Elastic Strain Energy
Section 8. 8. Elastic Strain Energy The strain energy stored in an elastic material upon deformation is calculated below for a number of different geometries and loading conditions. These expressions for
More informationProjectile motion simulator. http://www.walterfendt.de/ph11e/projectile.htm
More Chapter 3 Projectile motion simulator http://www.walterfendt.de/ph11e/projectile.htm The equations of motion for constant acceleration from chapter 2 are valid separately for both motion in the x
More informationThe dynamic equation for the angular motion of the wheel is R w F t R w F w ]/ J w
Chapter 4 Vehicle Dynamics 4.. Introduction In order to design a controller, a good representative model of the system is needed. A vehicle mathematical model, which is appropriate for both acceleration
More informationAn Introduction to Applied Mathematics: An Iterative Process
An Introduction to Applied Mathematics: An Iterative Process Applied mathematics seeks to make predictions about some topic such as weather prediction, future value of an investment, the speed of a falling
More informationChapter 2. Mission Analysis. 2.1 Mission Geometry
Chapter 2 Mission Analysis As noted in Chapter 1, orbital and attitude dynamics must be considered as coupled. That is to say, the orbital motion of a spacecraft affects the attitude motion, and the attitude
More informationExample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x
Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written
More informationPhysics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE
1 P a g e Motion Physics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE If an object changes its position with respect to its surroundings with time, then it is called in motion. Rest If an object
More information