Control of Mobile Robots
|
|
- Suzan Newton
- 5 years ago
- Views:
Transcription
1 Control of Mobile Robots Regulation and trajectory tracking Prof. Luca Bascetta Politecnico di Milano Dipartimento di Elettronica, Informazione e Bioingegneria
2 Organization and motivations 2 We complete the navigation part introducing the control problem and control techniques for mobile robots. The main topics on regulation and trajectory tracking are control review differentially flat systems exact linearization a trajectory tracking controller for a unicycle robot a regulation controller for a unicycle robot fundamental of odometric localization
3 Control review Lyapunov direct method 3 We start reviewing a fundamental tool to prove the stability of nonlinear control systems. Consider the following nonlinear time invariant system assuming ff 00 = 00, xx = 00 is an equilibrium state. We would like to introduce a tool to demonstrate the global asymptotic stability of the equilibrium of the nonlinear invariant system. The Lyapunov direct method is based on the following idea: if we can associate an energy-based description to an autonomous dynamic system, and for each system state, except for the equilibrium, the time derivative of the energy function is negative, then energy decreases along any system trajectory until it reaches its minimum at the equilibrium state this justifies an intuitive concept of stability.
4 Control review Lyapunov direct method 4 We introduce this energy-based function, as a scalar function VV xx of the system state, continuous together with its first derivative. VV xx is a Lyapunov function if The existence of such function ensures the global asymptotic stability of the equilibrium xx = 00. The equilibrium is globally asymptotically stable if a positive definite, radially unbounded function VV is found so that its time derivative along the system trajectories is negative definite.
5 Control review Lyapunov direct method 5 Let s introduce a quadratic form the time derivative is If ff xx renders the function VV xx negative definite, VV xx is a Lyapuov function and xx = 0 is globally asymptotically stable. If VV xx is not negative definite, nothing can be inferred on the stability of the system. One should resort to different choices of VV xx in order to find, if it is possible, a negative definite VV xx. The Lyapunov direct method gives only a sufficient condition.
6 Control review Lyapunov direct method (example) 6 Consider the equations of a pendulum with friction Let s introduce the following candidate Lyapunov function whose time derivative along system trajectories is VV xx is negative semidefinite but not negative definite, in fact VV xx = 0 for xx 2 = 0 irrespective of the value of xx 1.
7 Control review Lyapunov direct method (example) 7 Let s consider the physical system. VV xx is negative everywhere except on the line xx 2 = 0. According to the system dynamics, however assuming ππ < xx 1 < ππ, VV xx = 00 only for xx 1 = xx 2 = 0. Consequently, VV xx decreases toward 0 and, consequently, xx tt 0 as tt. This is consistent with the physical understanding that, due to friction, energy cannot remain constant while the system is in motion.
8 Control review Lyapunov direct method, La Salle s invariance principle 8 We can formalize the previous result as follows: if in a domain about the origin we can find a Lyapunov function whose derivative along the trajectories of the system is negative semidefinite, and if we can establish that no trajectory can stay identically at points where VV xx = 00 except at the origin, then the origin is globally asymptotically stable. This idea follows from La Salle s invariance principle. We still have one question to solve: La Salle s invariance principle does not hold for time varying systems, how to assess the stability of
9 Control review Lyapunov direct method, Barbalat lemma 9 Given a scalar function VV xx, tt so that VV xx, tt is lower bounded VV xx, tt 0 VV xx, tt is uniformly continuous then lim VV xx, tt = 0. tt 0 The last condition, that is difficult to verify, can be substituted by VV xx, tt is bounded Barbalat lemma can be used for time invariant systems as well, as an alternative to La Salle s invariance principle.
10 Differentially flat systems 10 We start introducing properties and tools that allow to transform a nonlinear dynamic system in a simpler form. The first property we introduce is called differential flatness. A nonlinear system is differentially flat if there exists a set of outputs yy, called flat outputs, such that the state xx and the control inputs uu can be expressed algebraically as a function of yy and its time derivatives up to a certain order
11 Differentially flat systems: the unicycle example 11 A unicycle kinematic model is a differentially flat system, when flat outputs zz 1 = xx and zz 2 = yy are selected. The state vector of the kinematic model can be expressed as a function of the flat outputs and their first derivatives
12 Differentially flat systems: the unicycle example 12 The input vector of the kinematic model can be expressed as a function of the flat outputs and their first and second order derivatives. Using the first two equations of the model we can derive a relation for the linear velocity (assuming vv > 0) From the third equation of the model and the flat relation on θθ we derive a relation for the angular velocity
13 Differentially flat systems: the unicycle example 13 Summarizing, the mappings obtained by the flat transformation are and We must observe that this mapping is singular when In this case the linear velocity is zero, and θθ and ωω are not well-defined.
14 Differentially flat systems: the bicycle example 14 A bicycle kinematic model is a differentially flat system, when flat outputs zz 1 = xx and zz 2 = yy are selected. The state vector of the kinematic model can be expressed as a function of the flat outputs and their first derivatives, as for the unicycle model
15 Differentially flat systems: the bicycle example 15 From the first two equations we get also and differentiating the first relation with respect to time From the last equation we now obtain
16 Differentially flat systems: the bicycle example 16 As for the input vector, the linear velocity can be obtained from the first two equations, as in the unicycle case (again assuming vv > 0) The second input can be instead derived from the last equation
17 Differentially flat systems 17 Once the states and inputs have been expressed in terms of the flat outputs, given an output trajectory yy(tt), the associated trajectory of the state xx and of the control inputs uu are uniquely determined.
18 Exact linearization of a nonlinear system: the unicycle kinematic model 18 The kinematic and dynamic models of mobile robots we have introduced so far are always represented by nonlinear dynamic systems. We now introduce a change of coordinates that transforms a nonlinear system into a linear one. We start considering the example of a unicycle model differentiating the first two equations with respect to time
19 Exact linearization of a nonlinear system: the unicycle kinematic model 19 Multiplying the two equations by cos θθ / sin θθ and summing them together Multiplying the two equations by sin θθ / cos θθ and subtracting them together
20 Exact linearization of a nonlinear system: the unicycle kinematic model 20 Summarizing, we found the two change of coordinates
21 Exact linearization of a nonlinear system: the unicycle kinematic model 21 Some remarks: the change of variables we have introduced is called feedback linearization as a result of the change of variables the closed-loop system (from aa xx to xx, or from aa yy to yy) is described by two independent double integrators the change of variables has a singularity when vv = 0 the change of variables is dynamic Can we introduce a different change of variables that is not affected by this singularity?
22 Exact linearization of a nonlinear system: the unicycle kinematic model wrt point P 22 Consider again the unicycle model, making reference to the motion of point PP and differentiating with respect to time We can now apply the same procedure used to derive the previous change of variable
23 Exact linearization of a nonlinear system: the unicycle kinematic model wrt point P 23 Multiplying the two equations by cos θθ / sin θθ and summing them together Multiplying the two equations by sin θθ / cos θθ and subtracting them together
24 Exact linearization of a nonlinear system: the unicycle kinematic model wrt point P 24 Summarizing, we found the two change of coordinates
25 Exact linearization of a nonlinear system: the unicycle kinematic model wrt point P 25 Some remarks: the change of coordinates has no more singularities the change of coordinates is static the closed-loop system is described by two independent integrators, it is thus more simple especially from a control point of view an outer control loop should regulate the position of point PP Can we use these changes of coordinates to linearize the bicycle kinematic model?
26 Exact linearization of a nonlinear system: the bicycle kinematic model 26 Consider a simplified version of the rear-wheel drive bicycle model We can apply a similar change of coordinates, considering that ωω = vv tan φφ. l Considering, for example, the linearization with respect to point PP, we have Let s concentrate on the second equation
27 Exact linearization of a nonlinear system: the bicycle kinematic model 27 Summarizing, the change of coordinates is and the closed-loop system is again characterized by two independent integrators But again the change of coordinates is singular when vv = 0.
28 Exact linearization of the unicycle an bicycle models: an interpretation 28 Before studying the linearization of a dynamic model, we would like to investigate more the exact linearization tool. In all the examples we started from a 3D configuration space, representing robot pose, and we end up with a reduced configuration space, representing robot position. From a physical point of view, the model in the new coordinates describes the motion of the robot as the motion of a particle. This particle can move in the 2D space and it is not subjected to any constraint. From a system theory point of view, the change of coordinates transforms a 3 rd order system into a 2 nd order system, it is thus a feedback that induces a loss of observability. For a robot model the heading is no more observable from the output.
29 Exact linearization of the unicycle an bicycle models: an interpretation 29 What are the main consequences? The linearizing feedback induces a hidden dynamics that can/cannot be asymptotically stable if it is not asymptotically stable the change of coordinates cannot be applied The heading is no more observable, an outer controller cannot control the heading of the robot Let s consider an example
30 Exact linearization of the unicycle an bicycle models: an interpretation 30 Consider the unicycle model linearized with respect to a point PP and the change of coordinates The closed-loop system is described by the following dynamic system
31 Exact linearization of the unicycle an bicycle models: an interpretation 31 We assume that PP moves at constant velocity along a straight line with vv PP, θθ PP constant and vv PP > 0. The equations of the closed-loop system become Focusing on the unobservable state, we define Δθθ = θθ θθ PP.
32 Exact linearization of the unicycle an bicycle models: an interpretation 32 The heading dynamics can be written as Unstable Asymptotically stable and the equilibria of this nonlinear system are θθ = θθ PP + 2kkkk, kk Z, asymptotically stable with a basin of attraction ( θθ PP + 2(kk
33 Exact linearization of a nonlinear system: the bicycle dynamic model 33 What about linearizing a dynamic model? We consider the example of the bicycle model As for the unicycle kinematic model we can linearize with respect to xx, yy or with respect to a point PP that does not belong to the vehicle. We consider the linearization with respect to a point PP.
34 Exact linearization of a nonlinear system: the bicycle dynamic model 34 Consider a point PP at a distance E from the vehicle CoG along the velocity vector and differentiating with respect to time
35 Exact linearization of a nonlinear system: the bicycle dynamic model 35 Introducing the sideslip dynamics in the previous equations we obtain the change of coordinates
36 Exact linearization of a nonlinear system: the bicycle dynamic model 36 Finally, from the definition of ωω we obtain Summarizing, if we apply the change of coordinates the bicycle dynamic model is transformed into two independent integrators
37 Exact linearization of a nonlinear system: the bicycle dynamic model 37 Some remarks: as for the kinematic model, the linearized system represent the motion of a particle sideslip and yaw dynamics are hidden by the change of coordinates the change of coordinates is static but is singular when VV = 0 the change of coordinates is now a function of the dynamic model parameters (MM, CC ff, CC rr, aa, bb), there can be robustness issues
38 Exact linearization of a nonlinear system: generic dynamic model 38 The generic expression of the dynamic model of a mobile robot we have introduced allows to devise a general but partial feedback linearization. Assuming a control availability assumption that is often satisfied, we can select where aa is the new control variable. The closed-loop system is reduced to mm independent integrators Kinematic model
39 Exact linearization of a nonlinear system: generic dynamic model 39 Some remarks: it is a general approach similar to the one used to linearize the model of a manipulator as for the previous linearization of dynamic models, the linearizing law entails model parameters, a robustness issue can be thus arise if the system is unconstrained and fully actuated the linearizing law reduces to and it is thus equivalent to an inverse dynamics control. The linearized system is equivalent to nn decoupled double integrators
40 The motion control problem 40 The motion control problem for a mobile robot can be formulated with respect to kinematic model dynamic model At least two reasons allow to go for the first option: Dynamic effects are handled by low level control systems or can be neglected (low velocities/accelerations) Dynamic effects are more important in autonomous vehicles than in mobile robotics dynamics can be cancelled out with a dynamic state feedback in the majority of the robots wheel torques cannot be accessed, as there are low level control loops integrated in the hardware architecture, and generalized velocities are usually the only accessible commands
41 The motion control problem 41 Two different control problems can be considered: trajectory or path tracking, the robot must asymptotically track a desired Cartesian path or trajectory posture regulation, the robot must asymptotically reach a given posture A planning step is not required, but the Cartesian trajectory of the robot cannot be predicted Forcing the robot to move along or close to a trajectory planned in advanced considerably reduces the risk of collisions
42 Deriving a trajectory tracking controller for the unicycle model 42 Let s assume that the desired trajectory xx dd tt, yy dd tt satisfies the unicycle kinematic model. Exploiting the flatness property we can compute the desired values for the angle θθ dd tt and for the linear and angular velocities vv dd tt and ωω dd tt and The role of a trajectory tracking controller is to ensure that the tracking error asymptotically vanishes. We thus need to define the tracking error
43 Deriving a trajectory tracking controller for the unicycle model 43 We could define the tracking error as but it is more convenient to project the positional part of the error onto a frame that has the same orientation of the robot local reference frame Let s compute the time derivative of these equations
44 Deriving a trajectory tracking controller for the unicycle model 44 Let s compute the time derivative of the first equation it follows Applying the same procedure to the second equation we obtain Finally
45 Deriving a trajectory tracking controller for the unicycle model 45 Summarizing Introducing now a change of variables on the inputs In general it is a time varying system We obtain linear term nonlinear terms
46 Deriving a trajectory tracking controller for the unicycle model 46 Starting from the system that describes the error dynamics we can derive two trajectory tracking controllers a linear controller based on the linearized error dynamics a nonlinear controller
47 A linear trajectory tracking controller for the unicycle model 47 A simple way to derive a linear controller, is by linearizing the error dynamics along the reference trajectory (ee = 00) we obtain a linear but time varying system. Introducing the control law we get a closed loop error dynamics described by the following linear time varying system
48 A linear trajectory tracking controller for the unicycle model 48 The characteristic polynomial of matrix AA tt is Assuming kk 1 = kk 3 we can rewrite the polynomial in this way The closed loop linearized error dynamics is characterized by a real pole kk 1 two real or complex poles
49 A linear trajectory tracking controller for the unicycle model 49 Let s assume that the second order polynomial is characterized by complex poles with natural frequency ωω nn and damping ξξ Summarizing, the closed loop linearized error dynamics is characterized by one real negative eigenvalue in 2ξξωω nn two complex eigenvalues with damping ξξ and natural frequency ωω nn
50 A linear trajectory tracking controller for the unicycle model 50 A few remarks: in general the closed loop linearized error dynamics is a time varying system, there is thus no guarantee that the system is asymptotically stable if vv dd and ωω dd are constant (circular and rectilinear trajectories), the linearized system is time invariant, and the choice of gains ensures it is asymptotically stabile. As a consequence, with the control law the origin of the original nonlinear error system is asymptotically stable (not globally!). For sufficiently small initial errors the unicycle converges to the desired trajectory.
51 A linear trajectory tracking controller for the unicycle model 51 in general the control law we have defined is linear but time varying the robot commands vv and ωω are given by the angular velocity ωω diverges when vv dd 0, this control law can be used only for persistent Cartesian trajectories for which with this control law motion inversions on the reference trajectory are not allowed
52 A nonlinear trajectory tracking controller for the unicycle model 52 Let s go back to the nonlinear error dynamics substituting the input transformation ωω = ωω dd uu 2 we obtain the nonlinear version of the previous control law
53 A nonlinear trajectory tracking controller for the unicycle model 53 where kk 1 vv dd, ωω dd > 0 and kk 3 vv dd, ωω dd > 0 are bounded functions with bounded derivatives kk 2 > 0 is constant If reference inputs vv dd and ωω dd are bounded with bounded derivatives do not both converge to zero the tracking error ee globally converges to zero (for any initial condition!).
54 A nonlinear trajectory tracking controller for the unicycle model 54 The corresponding velocity commands are given by Let s prove that the tracking error converges to zero globally.
55 A nonlinear trajectory tracking controller for the unicycle model 55 First of all the closed loop error dynamics are given by We consider the candidate Lyapunov function and its time derivative along the system trajectories that is negative semi-definite. As the system is time varying we have to use Barbalat lemma.
56 A nonlinear trajectory tracking controller for the unicycle model 56 Let s consider we have that VV xx, tt is lower bounded and ee is bounded VV xx, tt 0 VV xx, tt is uniformly continuous (or VV xx, tt is bounded) The system is time varying, we cannot use La Salle invariance principle and using Barbalat lemma we conclude that lim VV xx, tt = 0. This implies that ee 1 and ee 3 tt 0 tend to 0.
57 A nonlinear trajectory tracking controller for the unicycle model 57 Considering that ee 1 and ee 3 tend to zero, from system equations it follows that and thus ee 2 tends to zero as well, provided that at least one of the reference inputs is persistent.
58 A feedback linearization based trajectory tracking controller for the unicycle model 58 Remember that, making reference to the motion of point PP and introducing the feedback linearizing law we obtain The dynamics of point PP is now linear and can be controlled with a linear regulator
59 A feedback linearization based trajectory tracking controller for the unicycle model 59 A simple linear controller with kk 1, kk 2 > 0, guarantees exponential convergence to zero of the Cartesian tracking error with decoupled dynamics The orientation of the robot is not controlled and evolves according to but at least at steady state the equilibrium of this system is
60 Deriving a regulation controller for the unicycle model 60 The regulation problem is the problem of designing a feedback control law that drives the robot to a desired configuration qq dd (and stop there). None of the tracking controllers allow to solve this problem: the controller based on approximate linearization requires a persistent state trajectory the nonlinear controller requires a persistent state trajectory the feedback linearization based controller does not allow to control the final orientation at the destination point For these reasons, we now introduce two control laws for Cartesian and posture regulation.
61 A Cartesian regulation controller for the unicycle model 61 Consider the problem of controlling only the final position of the robot, without specifying the final orientation. Assume that the desired Cartesian position is the origin, the error is We can introduce the following control law Cartesian error projected on the sagittal axis where kk 1, kk 2 > 0. Let s analyze the stability of the closed loop system. Pointing error
62 A Cartesian regulation controller for the unicycle model 62 Consider the Lyapunov-like function that is only positive semi-definite at the origin (it is zero in all configurations such that xx = yy = 0, independently from the value of θθ). The time derivative along the system trajectories is that is negative semi-definite at the origin. We conclude that: VV xx, tt is lower bounded and ee PP is bounded VV xx, tt is uniformly continuous (or and Barbalat lemma implies that lim tt VV xx, tt is bounded) VV xx, tt = 0.
63 A Cartesian regulation controller for the unicycle model 63 From lim VV xx, tt = 0, it follows tt and thus the projection of the Cartesian error ee PP on the sagittal axis of the unicycle vanishes. This can happen only at the origin, otherwise the steering velocity would force the unicycle to rotate so as to align with ee PP. We can conclude that the Cartesian error tends to zero for any initial configuration.
64 A posture regulation controller for the unicycle model 64 We assume again that the desired configuration is the origin qq dd = TT. We introduce the following variables to express the unicycle model in polar coordinates Let s start from the first equation
65 A posture regulation controller for the unicycle model 65 Summarizing Moving now to the second equation we get
66 A posture regulation controller for the unicycle model 66 Finally, for the last equation we get Summarizing, the unicycle model in polar coordinates is described by Note that the model is singular for ρρ = 0
67 A posture regulation controller for the unicycle model 67 We introduce the following feedback control law with kk 1, kk 2 > 0. Let s prove the asymptotic convergence of the closed loop system to the desired configuration qq dd.
68 A posture regulation controller for the unicycle model 68 Let s consider the following Lyapunov candidate The time derivative along the system trajectories is and is negative semi-definite.
69 A posture regulation controller for the unicycle model 69 We conclude that: VV xx, tt is lower bounded and the norm of the state is bounded VV xx, tt is uniformly continuous (or VV xx, tt is bounded) and Barbalat lemma implies that lim VV xx, tt = 0. tt As a consequence, ρρ and γγ tend to zero and and thus, to guarantee that qq dd is an equilibrium point, δδ must tend to zero.
70 Odometric localization 70 As we already know, to control a mobile robot we need to estimate in real-time the robot configuration at each time instant. This is called localization problem. Consider a unicycle robot moving under the action of velocity commands vv and ωω, constant within each sampling interval. In the sampling interval the robot moves along an arc of circle of radius RR = vv kk ωω kk a line segment if ωω kk = 0 Assume the robot configuration at time tt kk is known, qq tt kk vv kk and ωω kk in the interval [tt kk, tt kk+1 ). = qq kk, together with the inputs
71 Odometric localization 71 Using forward integration of the kinematic model with the Euler method, we can derive the configuration qq kk+1 at time tt kk+1 where TT ss = tt kk+1 tt kk. These relations are approximated as they assume θθ kk constant in the integration period This relation is exact
72 Odometric localization 72 A more accurate estimate can be achieved adopting the second-order Runge-Kutta integration method More complex and accurate methods can be devised. The average value of the orientation in the integration period We have now to relate our estimators to a set of available measurements, wheel encoder measurements.
73 Odometric localization 73 Consider, for example, the case of a differential drive robot. If Δφφ RR and Δφφ LL are the rotations of the wheels measured by the incremental encoders during a sampling interval The linear and angular velocities can be estimated as The method we have introduced is called odometric localization or passive localization or dead reckoning. This method is subject to an error that grows over time (drift) that quickly becomes significant over sufficiently long paths. This is due to several causes, including wheel slippage, inaccuracy in kinematic parameters, numerical errors in the integration,
Automatic Control Systems theory overview (discrete time systems)
Automatic Control Systems theory overview (discrete time systems) Prof. Luca Bascetta (luca.bascetta@polimi.it) Politecnico di Milano Dipartimento di Elettronica, Informazione e Bioingegneria Motivations
More informationControl of Robotic Manipulators
Control of Robotic Manipulators Set Point Control Technique 1: Joint PD Control Joint torque Joint position error Joint velocity error Why 0? Equivalent to adding a virtual spring and damper to the joints
More informationControl of Mobile Robots Prof. Luca Bascetta
Control of Mobile Robots Prof. Luca Bascetta EXERCISE 1 1. Consider a wheel rolling without slipping on the horizontal plane, keeping the sagittal plane in the vertical direction. Write the expression
More informationEN Nonlinear Control and Planning in Robotics Lecture 3: Stability February 4, 2015
EN530.678 Nonlinear Control and Planning in Robotics Lecture 3: Stability February 4, 2015 Prof: Marin Kobilarov 0.1 Model prerequisites Consider ẋ = f(t, x). We will make the following basic assumptions
More informationRobot Control Basics CS 685
Robot Control Basics CS 685 Control basics Use some concepts from control theory to understand and learn how to control robots Control Theory general field studies control and understanding of behavior
More informationSECTION 7: STEADY-STATE ERROR. ESE 499 Feedback Control Systems
SECTION 7: STEADY-STATE ERROR ESE 499 Feedback Control Systems 2 Introduction Steady-State Error Introduction 3 Consider a simple unity-feedback system The error is the difference between the reference
More informationVideo 8.1 Vijay Kumar. Property of University of Pennsylvania, Vijay Kumar
Video 8.1 Vijay Kumar 1 Definitions State State equations Equilibrium 2 Stability Stable Unstable Neutrally (Critically) Stable 3 Stability Translate the origin to x e x(t) =0 is stable (Lyapunov stable)
More information3. Mathematical Modelling
3. Mathematical Modelling 3.1 Modelling principles 3.1.1 Model types 3.1.2 Model construction 3.1.3 Modelling from first principles 3.2 Models for technical systems 3.2.1 Electrical systems 3.2.2 Mechanical
More informationAngular Momentum, Electromagnetic Waves
Angular Momentum, Electromagnetic Waves Lecture33: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay As before, we keep in view the four Maxwell s equations for all our discussions.
More informationNonlinear System Analysis
Nonlinear System Analysis Lyapunov Based Approach Lecture 4 Module 1 Dr. Laxmidhar Behera Department of Electrical Engineering, Indian Institute of Technology, Kanpur. January 4, 2003 Intelligent Control
More informationTrajectory tracking & Path-following control
Cooperative Control of Multiple Robotic Vehicles: Theory and Practice Trajectory tracking & Path-following control EECI Graduate School on Control Supélec, Feb. 21-25, 2011 A word about T Tracking and
More informationPosture regulation for unicycle-like robots with. prescribed performance guarantees
Posture regulation for unicycle-like robots with prescribed performance guarantees Martina Zambelli, Yiannis Karayiannidis 2 and Dimos V. Dimarogonas ACCESS Linnaeus Center and Centre for Autonomous Systems,
More informationControl of industrial robots. Centralized control
Control of industrial robots Centralized control Prof. Paolo Rocco (paolo.rocco@polimi.it) Politecnico di Milano ipartimento di Elettronica, Informazione e Bioingegneria Introduction Centralized control
More informationChapter 3 Numerical Methods
Chapter 3 Numerical Methods Part 3 3.4 Differential Algebraic Systems 3.5 Integration of Differential Equations 1 Outline 3.4 Differential Algebraic Systems 3.4.1 Constrained Dynamics 3.4.2 First and Second
More informationSECTION 8: ROOT-LOCUS ANALYSIS. ESE 499 Feedback Control Systems
SECTION 8: ROOT-LOCUS ANALYSIS ESE 499 Feedback Control Systems 2 Introduction Introduction 3 Consider a general feedback system: Closed-loop transfer function is KKKK ss TT ss = 1 + KKKK ss HH ss GG ss
More informationA Posteriori Error Estimates For Discontinuous Galerkin Methods Using Non-polynomial Basis Functions
Lin Lin A Posteriori DG using Non-Polynomial Basis 1 A Posteriori Error Estimates For Discontinuous Galerkin Methods Using Non-polynomial Basis Functions Lin Lin Department of Mathematics, UC Berkeley;
More information(1) Introduction: a new basis set
() Introduction: a new basis set In scattering, we are solving the S eq. for arbitrary VV in integral form We look for solutions to unbound states: certain boundary conditions (EE > 0, plane and spherical
More informationRobot Manipulator Control. Hesheng Wang Dept. of Automation
Robot Manipulator Control Hesheng Wang Dept. of Automation Introduction Industrial robots work based on the teaching/playback scheme Operators teach the task procedure to a robot he robot plays back eecute
More informationLecture No. 1 Introduction to Method of Weighted Residuals. Solve the differential equation L (u) = p(x) in V where L is a differential operator
Lecture No. 1 Introduction to Method of Weighted Residuals Solve the differential equation L (u) = p(x) in V where L is a differential operator with boundary conditions S(u) = g(x) on Γ where S is a differential
More information1 Lyapunov theory of stability
M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 1 Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability
More informationSome examples of radical equations are. Unfortunately, the reverse implication does not hold for even numbers nn. We cannot
40 RD.5 Radical Equations In this section, we discuss techniques for solving radical equations. These are equations containing at least one radical expression with a variable, such as xx 2 = xx, or a variable
More informationSupport Vector Machines. CSE 4309 Machine Learning Vassilis Athitsos Computer Science and Engineering Department University of Texas at Arlington
Support Vector Machines CSE 4309 Machine Learning Vassilis Athitsos Computer Science and Engineering Department University of Texas at Arlington 1 A Linearly Separable Problem Consider the binary classification
More informationTopic # /31 Feedback Control Systems. Analysis of Nonlinear Systems Lyapunov Stability Analysis
Topic # 16.30/31 Feedback Control Systems Analysis of Nonlinear Systems Lyapunov Stability Analysis Fall 010 16.30/31 Lyapunov Stability Analysis Very general method to prove (or disprove) stability of
More informationRotational Motion. Chapter 10 of Essential University Physics, Richard Wolfson, 3 rd Edition
Rotational Motion Chapter 10 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 We ll look for a way to describe the combined (rotational) motion 2 Angle Measurements θθ ss rr rrrrrrrrrrrrrr
More informationControl of a Car-Like Vehicle with a Reference Model and Particularization
Control of a Car-Like Vehicle with a Reference Model and Particularization Luis Gracia Josep Tornero Department of Systems and Control Engineering Polytechnic University of Valencia Camino de Vera s/n,
More informationSecondary 3H Unit = 1 = 7. Lesson 3.3 Worksheet. Simplify: Lesson 3.6 Worksheet
Secondary H Unit Lesson Worksheet Simplify: mm + 2 mm 2 4 mm+6 mm + 2 mm 2 mm 20 mm+4 5 2 9+20 2 0+25 4 +2 2 + 2 8 2 6 5. 2 yy 2 + yy 6. +2 + 5 2 2 2 0 Lesson 6 Worksheet List all asymptotes, holes and
More informationMath 171 Spring 2017 Final Exam. Problem Worth
Math 171 Spring 2017 Final Exam Problem 1 2 3 4 5 6 7 8 9 10 11 Worth 9 6 6 5 9 8 5 8 8 8 10 12 13 14 15 16 17 18 19 20 21 22 Total 8 5 5 6 6 8 6 6 6 6 6 150 Last Name: First Name: Student ID: Section:
More informationPHY103A: Lecture # 4
Semester II, 2017-18 Department of Physics, IIT Kanpur PHY103A: Lecture # 4 (Text Book: Intro to Electrodynamics by Griffiths, 3 rd Ed.) Anand Kumar Jha 10-Jan-2018 Notes The Solutions to HW # 1 have been
More informationM.5 Modeling the Effect of Functional Responses
M.5 Modeling the Effect of Functional Responses The functional response is referred to the predation rate as a function of the number of prey per predator. It is recognized that as the number of prey increases,
More informationFeedback Control Strategies for a Nonholonomic Mobile Robot Using a Nonlinear Oscillator
Feedback Control Strategies for a Nonholonomic Mobile Robot Using a Nonlinear Oscillator Ranjan Mukherjee Department of Mechanical Engineering Michigan State University East Lansing, Michigan 4884 e-mail:
More informationRobust Control of Cooperative Underactuated Manipulators
Robust Control of Cooperative Underactuated Manipulators Marcel Bergerman * Yangsheng Xu +,** Yun-Hui Liu ** * Automation Institute Informatics Technology Center Campinas SP Brazil + The Robotics Institute
More informationKeywords : H control, robust control, wheeled mobile robots, uncertain systems, Disturbance rejection. GJRE-F Classification: FOR Code: p
Global Journal of researches in engineering Electrical and electronics engineering Volume 12 Issue 1 Version 1. January 212 Type: Double Blind Peer Reviewed International Research Journal Publisher: Global
More information10.4 Controller synthesis using discrete-time model Example: comparison of various controllers
10. Digital 10.1 Basic principle of digital control 10.2 Digital PID controllers 10.2.1 A 2DOF continuous-time PID controller 10.2.2 Discretisation of PID controllers 10.2.3 Implementation and tuning 10.3
More informationLecture 6. Notes on Linear Algebra. Perceptron
Lecture 6. Notes on Linear Algebra. Perceptron COMP90051 Statistical Machine Learning Semester 2, 2017 Lecturer: Andrey Kan Copyright: University of Melbourne This lecture Notes on linear algebra Vectors
More informationPhoton Interactions in Matter
Radiation Dosimetry Attix 7 Photon Interactions in Matter Ho Kyung Kim hokyung@pusan.ac.kr Pusan National University References F. H. Attix, Introduction to Radiological Physics and Radiation Dosimetry,
More informationModule 7 (Lecture 25) RETAINING WALLS
Module 7 (Lecture 25) RETAINING WALLS Topics Check for Bearing Capacity Failure Example Factor of Safety Against Overturning Factor of Safety Against Sliding Factor of Safety Against Bearing Capacity Failure
More information10.1 Three Dimensional Space
Math 172 Chapter 10A notes Page 1 of 12 10.1 Three Dimensional Space 2D space 0 xx.. xx-, 0 yy yy-, PP(xx, yy) [Fig. 1] Point PP represented by (xx, yy), an ordered pair of real nos. Set of all ordered
More informationGrover s algorithm. We want to find aa. Search in an unordered database. QC oracle (as usual) Usual trick
Grover s algorithm Search in an unordered database Example: phonebook, need to find a person from a phone number Actually, something else, like hard (e.g., NP-complete) problem 0, xx aa Black box ff xx
More informationMathematics Ext 2. HSC 2014 Solutions. Suite 403, 410 Elizabeth St, Surry Hills NSW 2010 keystoneeducation.com.
Mathematics Ext HSC 4 Solutions Suite 43, 4 Elizabeth St, Surry Hills NSW info@keystoneeducation.com.au keystoneeducation.com.au Mathematics Extension : HSC 4 Solutions Contents Multiple Choice... 3 Question...
More informationGeorgia Institute of Technology Nonlinear Controls Theory Primer ME 6402
Georgia Institute of Technology Nonlinear Controls Theory Primer ME 640 Ajeya Karajgikar April 6, 011 Definition Stability (Lyapunov): The equilibrium state x = 0 is said to be stable if, for any R > 0,
More informationCHAPTER 1. Introduction
CHAPTER 1 Introduction Linear geometric control theory was initiated in the beginning of the 1970 s, see for example, [1, 7]. A good summary of the subject is the book by Wonham [17]. The term geometric
More informationCDS 101/110: Lecture 3.1 Linear Systems
CDS /: Lecture 3. Linear Systems Goals for Today: Describe and motivate linear system models: Summarize properties, examples, and tools Joel Burdick (substituting for Richard Murray) jwb@robotics.caltech.edu,
More informationNONLINEAR PATH CONTROL FOR A DIFFERENTIAL DRIVE MOBILE ROBOT
NONLINEAR PATH CONTROL FOR A DIFFERENTIAL DRIVE MOBILE ROBOT Plamen PETROV Lubomir DIMITROV Technical University of Sofia Bulgaria Abstract. A nonlinear feedback path controller for a differential drive
More informationADAPTIVE NEURAL NETWORK CONTROLLER DESIGN FOR BLENDED-WING UAV WITH COMPLEX DAMAGE
ADAPTIVE NEURAL NETWORK CONTROLLER DESIGN FOR BLENDED-WING UAV WITH COMPLEX DAMAGE Kijoon Kim*, Jongmin Ahn**, Seungkeun Kim*, Jinyoung Suk* *Chungnam National University, **Agency for Defense and Development
More informationWork, Energy, and Power. Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition
Work, Energy, and Power Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 With the knowledge we got so far, we can handle the situation on the left but not the one on the right.
More informationCURVATURE BASED POINT STABILIZATION AND PATH FOLLOWING FOR COMPLIANT FRAMED WHEELED MODULAR MOBILE ROBOTS. Brian W. Albiston.
, CURVATURE BASED POINT STABILIZATION AND PATH FOLLOWING FOR COMPLIANT FRAMED WHEELED MODULAR MOBILE ROBOTS by Brian W. Albiston A thesis submitted to the faculty of The University of Utah in partial fulfillment
More informationQuaternion-Based Tracking Control Law Design For Tracking Mode
A. M. Elbeltagy Egyptian Armed forces Conference on small satellites. 2016 Logan, Utah, USA Paper objectives Introduction Presentation Agenda Spacecraft combined nonlinear model Proposed RW nonlinear attitude
More informationSECTION 7: FAULT ANALYSIS. ESE 470 Energy Distribution Systems
SECTION 7: FAULT ANALYSIS ESE 470 Energy Distribution Systems 2 Introduction Power System Faults 3 Faults in three-phase power systems are short circuits Line-to-ground Line-to-line Result in the flow
More informationKINEMATICS REVIEW VECTOR ALGEBRA - SUMMARY
1 KINEMATICS REVIEW VECTOR ALGEBRA - SUMMARY Magnitude A numerical value with appropriate units. Scalar is a quantity that is completely specified by magnitude. Vector requires both, magnitude and direction
More informationPHY103A: Lecture # 9
Semester II, 2017-18 Department of Physics, IIT Kanpur PHY103A: Lecture # 9 (Text Book: Intro to Electrodynamics by Griffiths, 3 rd Ed.) Anand Kumar Jha 20-Jan-2018 Summary of Lecture # 8: Force per unit
More informationRepresent this system in terms of a block diagram consisting only of. g From Newton s law: 2 : θ sin θ 9 θ ` T
Exercise (Block diagram decomposition). Consider a system P that maps each input to the solutions of 9 4 ` 3 9 Represent this system in terms of a block diagram consisting only of integrator systems, represented
More informationCDS 101 Precourse Phase Plane Analysis and Stability
CDS 101 Precourse Phase Plane Analysis and Stability Melvin Leok Control and Dynamical Systems California Institute of Technology Pasadena, CA, 26 September, 2002. mleok@cds.caltech.edu http://www.cds.caltech.edu/
More informationCharge carrier density in metals and semiconductors
Charge carrier density in metals and semiconductors 1. Introduction The Hall Effect Particles must overlap for the permutation symmetry to be relevant. We saw examples of this in the exchange energy in
More informationQuantum Mechanics. An essential theory to understand properties of matter and light. Chemical Electronic Magnetic Thermal Optical Etc.
Quantum Mechanics An essential theory to understand properties of matter and light. Chemical Electronic Magnetic Thermal Optical Etc. Fall 2018 Prof. Sergio B. Mendes 1 CHAPTER 3 Experimental Basis of
More informationSECTION 5: POWER FLOW. ESE 470 Energy Distribution Systems
SECTION 5: POWER FLOW ESE 470 Energy Distribution Systems 2 Introduction Nodal Analysis 3 Consider the following circuit Three voltage sources VV sss, VV sss, VV sss Generic branch impedances Could be
More informationAnalysis and Design of Control Dynamics of Manipulator Robot s Joint Drive
Journal of Mechanics Engineering and Automation 8 (2018) 205-213 doi: 10.17265/2159-5275/2018.05.003 D DAVID PUBLISHING Analysis and Design of Control Dynamics of Manipulator Robot s Joint Drive Bukhar
More information10.4 The Cross Product
Math 172 Chapter 10B notes Page 1 of 9 10.4 The Cross Product The cross product, or vector product, is defined in 3 dimensions only. Let aa = aa 1, aa 2, aa 3 bb = bb 1, bb 2, bb 3 then aa bb = aa 2 bb
More informationCONTROL OF THE NONHOLONOMIC INTEGRATOR
June 6, 25 CONTROL OF THE NONHOLONOMIC INTEGRATOR R. N. Banavar (Work done with V. Sankaranarayanan) Systems & Control Engg. Indian Institute of Technology, Bombay Mumbai -INDIA. banavar@iitb.ac.in Outline
More information(W: 12:05-1:50, 50-N202)
2016 School of Information Technology and Electrical Engineering at the University of Queensland Schedule of Events Week Date Lecture (W: 12:05-1:50, 50-N202) 1 27-Jul Introduction 2 Representing Position
More informationRobotics 1 Inverse kinematics
Robotics 1 Inverse kinematics Prof. Alessandro De Luca Robotics 1 1 Inverse kinematics what are we looking for? direct kinematics is always unique; how about inverse kinematics for this 6R robot? Robotics
More information4 Second-Order Systems
4 Second-Order Systems Second-order autonomous systems occupy an important place in the study of nonlinear systems because solution trajectories can be represented in the plane. This allows for easy visualization
More informationWave Motion. Chapter 14 of Essential University Physics, Richard Wolfson, 3 rd Edition
Wave Motion Chapter 14 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 Waves: propagation of energy, not particles 2 Longitudinal Waves: disturbance is along the direction of wave propagation
More informationDynamical Systems & Lyapunov Stability
Dynamical Systems & Lyapunov Stability Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline Ordinary Differential Equations Existence & uniqueness Continuous dependence
More informationIntegrating Rational functions by the Method of Partial fraction Decomposition. Antony L. Foster
Integrating Rational functions by the Method of Partial fraction Decomposition By Antony L. Foster At times, especially in calculus, it is necessary, it is necessary to express a fraction as the sum of
More informationSpecialist Mathematics 2019 v1.2
Examination (15%) This sample has been compiled by the QCAA to assist and support teachers in planning and developing assessment instruments for individual school settings. The examination must ensure
More informationPhysics Circular Motion
FACULTY OF EDUCATION Department of Curriculum and Pedagogy Physics Circular Motion Science and Mathematics Education Research Group Supported by UBC Teaching and Learning Enhancement Fund 2012-2015 Question
More informationGravitation. Chapter 8 of Essential University Physics, Richard Wolfson, 3 rd Edition
Gravitation Chapter 8 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 What you are about to learn: Newton's law of universal gravitation About motion in circular and other orbits How to
More informationTTK4150 Nonlinear Control Systems Solution 6 Part 2
TTK4150 Nonlinear Control Systems Solution 6 Part 2 Department of Engineering Cybernetics Norwegian University of Science and Technology Fall 2003 Solution 1 Thesystemisgivenby φ = R (φ) ω and J 1 ω 1
More informationAutonomous Mobile Robot Design
Autonomous Mobile Robot Design Topic: Guidance and Control Introduction and PID Loops Dr. Kostas Alexis (CSE) Autonomous Robot Challenges How do I control where to go? Autonomous Mobile Robot Design Topic:
More informationTECHNICAL NOTE AUTOMATIC GENERATION OF POINT SPRING SUPPORTS BASED ON DEFINED SOIL PROFILES AND COLUMN-FOOTING PROPERTIES
COMPUTERS AND STRUCTURES, INC., FEBRUARY 2016 TECHNICAL NOTE AUTOMATIC GENERATION OF POINT SPRING SUPPORTS BASED ON DEFINED SOIL PROFILES AND COLUMN-FOOTING PROPERTIES Introduction This technical note
More informationAdaptive Robust Tracking Control of Robot Manipulators in the Task-space under Uncertainties
Australian Journal of Basic and Applied Sciences, 3(1): 308-322, 2009 ISSN 1991-8178 Adaptive Robust Tracking Control of Robot Manipulators in the Task-space under Uncertainties M.R.Soltanpour, M.M.Fateh
More informationTime Domain Analysis of Linear Systems Ch2. University of Central Oklahoma Dr. Mohamed Bingabr
Time Domain Analysis of Linear Systems Ch2 University of Central Oklahoma Dr. Mohamed Bingabr Outline Zero-input Response Impulse Response h(t) Convolution Zero-State Response System Stability System Response
More informationInverse differential kinematics Statics and force transformations
Robotics 1 Inverse differential kinematics Statics and force transformations Prof Alessandro De Luca Robotics 1 1 Inversion of differential kinematics! find the joint velocity vector that realizes a desired
More informationLecture Schedule Week Date Lecture (M: 2:05p-3:50, 50-N202)
J = x θ τ = J T F 2018 School of Information Technology and Electrical Engineering at the University of Queensland Lecture Schedule Week Date Lecture (M: 2:05p-3:50, 50-N202) 1 23-Jul Introduction + Representing
More informationRobot Dynamics II: Trajectories & Motion
Robot Dynamics II: Trajectories & Motion Are We There Yet? METR 4202: Advanced Control & Robotics Dr Surya Singh Lecture # 5 August 23, 2013 metr4202@itee.uq.edu.au http://itee.uq.edu.au/~metr4202/ 2013
More informationRational Equations and Graphs
RT.5 Rational Equations and Graphs Rational Equations In previous sections of this chapter, we worked with rational expressions. If two rational expressions are equated, a rational equation arises. Such
More informationAUTOMATIC CONTROL. Andrea M. Zanchettin, PhD Spring Semester, Introduction to Automatic Control & Linear systems (time domain)
1 AUTOMATIC CONTROL Andrea M. Zanchettin, PhD Spring Semester, 2018 Introduction to Automatic Control & Linear systems (time domain) 2 What is automatic control? From Wikipedia Control theory is an interdisciplinary
More informationControl of Electromechanical Systems
Control of Electromechanical Systems November 3, 27 Exercise Consider the feedback control scheme of the motor speed ω in Fig., where the torque actuation includes a time constant τ A =. s and a disturbance
More informationMathematical Theory of Control Systems Design
Mathematical Theory of Control Systems Design by V. N. Afarias'ev, V. B. Kolmanovskii and V. R. Nosov Moscow University of Electronics and Mathematics, Moscow, Russia KLUWER ACADEMIC PUBLISHERS DORDRECHT
More informationAttitude Regulation About a Fixed Rotation Axis
AIAA Journal of Guidance, Control, & Dynamics Revised Submission, December, 22 Attitude Regulation About a Fixed Rotation Axis Jonathan Lawton Raytheon Systems Inc. Tucson, Arizona 85734 Randal W. Beard
More informationAutonomous navigation of unicycle robots using MPC
Autonomous navigation of unicycle robots using MPC M. Farina marcello.farina@polimi.it Dipartimento di Elettronica e Informazione Politecnico di Milano 7 June 26 Outline Model and feedback linearization
More informationME5286 Robotics Spring 2017 Quiz 2
Page 1 of 5 ME5286 Robotics Spring 2017 Quiz 2 Total Points: 30 You are responsible for following these instructions. Please take a minute and read them completely. 1. Put your name on this page, any other
More informationRobotics, Geometry and Control - A Preview
Robotics, Geometry and Control - A Preview Ravi Banavar 1 1 Systems and Control Engineering IIT Bombay HYCON-EECI Graduate School - Spring 2008 Broad areas Types of manipulators - articulated mechanisms,
More informationF.3 Special Factoring and a General Strategy of Factoring
F.3 Special Factoring and a General Strategy of Factoring Difference of Squares section F4 233 Recall that in Section P2, we considered formulas that provide a shortcut for finding special products, such
More informationElio Sacco. Dipartimento di Ingegneria Civile e Meccanica Università di Cassino e LM
Elio Sacco Dipartimento di Ingegneria Civile e Meccanica Università di Cassino e LM The no-tension material model is adopted to evaluate the collapse load. y Thrust curve of the arch extrados intrados
More informationNonlinear Tracking Control of Underactuated Surface Vessel
American Control Conference June -. Portland OR USA FrB. Nonlinear Tracking Control of Underactuated Surface Vessel Wenjie Dong and Yi Guo Abstract We consider in this paper the tracking control problem
More informationKalman Filters for Mapping and Localization
Kalman Filters for Mapping and Localization Sensors If you can t model the world, then sensors are the robot s link to the external world (obsession with depth) Laser Kinect IR rangefinder sonar rangefinder
More informationReview for Exam Hyunse Yoon, Ph.D. Assistant Research Scientist IIHR-Hydroscience & Engineering University of Iowa
57:020 Fluids Mechanics Fall2013 1 Review for Exam3 12. 11. 2013 Hyunse Yoon, Ph.D. Assistant Research Scientist IIHR-Hydroscience & Engineering University of Iowa 57:020 Fluids Mechanics Fall2013 2 Chapter
More informationCoulomb s Law and Coulomb s Constant
Pre-Lab Quiz / PHYS 224 Coulomb s Law and Coulomb s Constant Your Name: Lab Section: 1. What will you investigate in this lab? 2. Consider a capacitor created when two identical conducting plates are placed
More information3. Fundamentals of Lyapunov Theory
Applied Nonlinear Control Nguyen an ien -.. Fundamentals of Lyapunov heory he objective of this chapter is to present Lyapunov stability theorem and illustrate its use in the analysis and the design of
More informationIntelligent Control. Module I- Neural Networks Lecture 7 Adaptive Learning Rate. Laxmidhar Behera
Intelligent Control Module I- Neural Networks Lecture 7 Adaptive Learning Rate Laxmidhar Behera Department of Electrical Engineering Indian Institute of Technology, Kanpur Recurrent Networks p.1/40 Subjects
More informationLecture 2: Plasma particles with E and B fields
Lecture 2: Plasma particles with E and B fields Today s Menu Magnetized plasma & Larmor radius Plasma s diamagnetism Charged particle in a multitude of EM fields: drift motion ExB drift, gradient drift,
More informationManipulator Dynamics (1) Read Chapter 6
Manipulator Dynamics (1) Read Capter 6 Wat is dynamics? Study te force (torque) required to cause te motion of robots just like engine power required to drive a automobile Most familiar formula: f = ma
More informationCOMPRESSION FOR QUANTUM POPULATION CODING
COMPRESSION FOR QUANTUM POPULATION CODING Ge Bai, The University of Hong Kong Collaborative work with: Yuxiang Yang, Giulio Chiribella, Masahito Hayashi INTRODUCTION Population: A group of identical states
More informationEML5311 Lyapunov Stability & Robust Control Design
EML5311 Lyapunov Stability & Robust Control Design 1 Lyapunov Stability criterion In Robust control design of nonlinear uncertain systems, stability theory plays an important role in engineering systems.
More informationLecture 22 Highlights Phys 402
Lecture 22 Highlights Phys 402 Scattering experiments are one of the most important ways to gain an understanding of the microscopic world that is described by quantum mechanics. The idea is to take a
More informationLinear-Quadratic Optimal Control: Full-State Feedback
Chapter 4 Linear-Quadratic Optimal Control: Full-State Feedback 1 Linear quadratic optimization is a basic method for designing controllers for linear (and often nonlinear) dynamical systems and is actually
More informationNon-Collision Conditions in Multi-agent Robots Formation using Local Potential Functions
2008 IEEE International Conference on Robotics and Automation Pasadena, CA, USA, May 19-23, 2008 Non-Collision Conditions in Multi-agent Robots Formation using Local Potential Functions E G Hernández-Martínez
More informationRigid-Body Attitude Control USING ROTATION MATRICES FOR CONTINUOUS, SINGULARITY-FREE CONTROL LAWS
Rigid-Body Attitude Control USING ROTATION MATRICES FOR CONTINUOUS, SINGULARITY-FREE CONTROL LAWS 3 IEEE CONTROL SYSTEMS MAGAZINE» JUNE -33X//$. IEEE SHANNON MASH Rigid-body attitude control is motivated
More informationAutomatic Control 2. Nonlinear systems. Prof. Alberto Bemporad. University of Trento. Academic year
Automatic Control 2 Nonlinear systems Prof. Alberto Bemporad University of Trento Academic year 2010-2011 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 1 / 18
More information