An Adaptive Attitude Control Formulation under Angular Velocity Constraints

Size: px
Start display at page:

Download "An Adaptive Attitude Control Formulation under Angular Velocity Constraints"

Transcription

1 An Adaptive Attitude Control Formulation under Angular Velocity Constraints Puneet Singla and Tarunraj Singh An adaptive attitude control law which explicitly takes into account the constraints on individual angular velocity components has been developed for rigid body attitude control problem. Rigorous stability analysis is presented in the paper which guarantee the asymptotic stability of the controller. The performance of the control laws for stable, bounded tracking of attitude trajectories is evaluated. The essential ideas and results from computer simulations are presented to illustrate the performance of the controller developed in this paper. I. Introduction Attitude control is the process of re-orienting a rigid body to a desired attitude or orientation and plays an important role in many applications ranging from various space and air transportation missions (autonomous mid-air re-fueling of an aircraft, International Space Station (ISS) supply and repair, and space systems automated inspection, servicing and assembly), to the control of robotic manipulators. Some of these applications such as mid-air aircraft refueling and space system automated inspection requires very precise rotational maneuvers. These requirements frequently necessitate the use of non-linear rigid body dynamic models for control system design. The attitude motion of a rigid body can be well represented by Euler s equations, 2 for nonlinear relative angular velocity evolution and attitude parameter kinematic equations. The rigid body attitude can be represented by many coordinate choices, 3, 4 but the quaternion representation is an ideal choice for the attitude estimation as it is free of geometrical singularities which is a desirable property when representing large angle amplitude trajectories. Although attitude kinematic and Euler s dynamic equations represent a near-exact dynamical model, for control design purposes, complications may arise from uncertain rigid body inertia which can change due to fuel consumption, solar array deployment, payload variation etc. Furthermore, stability robustness due to model errors and disturbances are primary consideration for design of any autonomous control system. Rigid body attitude control problems have been studied extensively in the literature. 5 Ref. 2 presents a very detailed review of earlier work for rigid-body attitude control problem. In Refs. 6, 8, 9 optimal attitude control laws are presented and in Refs. 5, 6,, 3 Lyapunov analysis based adaptive attitude tracking control Assistant Professor, AIAA, AAS Member, Department of Mechanical & Aerospace Engineering, University at Buffalo, Buffalo, NY-426, Professor, Department of Mechanical & Aerospace Engineering, University at Buffalo, Buffalo, NY-426, of 2

2 schemes are presented to compensate for the unknown rigid body inertia matrix. Although much progress has been made in rigid body attitude control in the presence of the rigid body inertia matrix uncertainties and external disturbances, there are no means of incorporating constraints on individual angular velocity. Constraints on rigid body angular velocity might be required for many applications such as rendezvous of space shuttle with ISS and mid-air refueling of an aircraft. The main objective of this paper is to develop an adaptive attitude control law to compensate for any errors in inertia matrix which takes into account the constraints on individual components of rigid body angular velocity. The adaptive control formulation in this paper is based upon Lyapunov s direct stability theorem and imposes the exact kinematic equations at the velocity level while taking care of model uncertainties at the acceleration level. An important contribution of the paper is the explicit consideration of constraints on rigid body angular velocity. The structure of this paper is as follows. First, the dynamical models for rigid body rotational motion is set forth followed by the development of adaptive attitude control law. Finally, the controllers designed in this paper are tested using numerical simulations. II. Attitude Dynamics In this section, we set forth the nonlinear rigid body dynamics model that we adopt for attitude kinematics and rotational dynamics. This model is described in order to be specific in the further developments contained in this paper. Rigid body kinematics can be represented by many coordinate choices, 3 but we prefer to use quaternion representation for attitude control since it is free of all geometrical singularities and has linear kinematic differential equations. The attitude motion of a rigid body is represented by nonlinear Euler s equations for angular velocity evolution and quaternion kinematic equations as given below: Quaternion Kinematics: q = 2 Ω(ω)q = 2 B(q)ω Euler s Equations: I ω = u ωiω (a) (b) where, q R 4 is a vector of quaternion that parameterize the rigid body attitude with respect to an inertial frame and ω R 3 represents the rigid body angular velocity expressed in the rigid body frame. u R 3 represents the vector of external torques. Further, I R 3 3 represents the rigid body inertia matrix, and 2 of 2

3 ω R 3 3 is a skew-symmetric matrix given as: ω = ω 3 ω 2 ω 3 ω ω 2 ω (2) Finally, Ω(ω) = ω 3 ω 2 ω ω 3 ω ω 2 ω 2 ω ω 3 ω ω 2 ω 3, B(q) = q 4 q 3 q 2 q 3 q 4 q q 2 q q 4 q q 2 q 3 (3a) III. Controller Formulation In this section, the velocity bounded adaptive control law will be derived for attitude control, using Lyapunov s direct stability theorem. The novel feature of the control law developed in this paper is that it explicitly accounts for bounds on uncertain rigid body inertia matrix and rigid body angular velocity. First, we will develop nominal attitude control for bounded rigid body angular velocity and later we will generalize the controller for bounds on uncertain rigid body inertia matrix. A. Feedback Control Formulation For Attitude Motion In this section, we seek to design a feedback control law for the system described by Eqs. (a) and (b) to regulate the rigid body attitude parameterized by reference quaternion q f such that ω (t) k, ω 2 (t) k 2, ω 3 (t) k 3, t (4) For this purpose, we define the error quaternion δq which represents the departure from the reference attitude trajectory q f. δq(t) = q(t) q f = q f 4 I q T f 3 q f3 q f3 q f4 q(t) (5) Making use of attitude kinematic Eq. (a), the attitude error kinematics can be written as: δ q = 2 Ωδq = B(δq)ω (6) 2 3 of 2

4 Now, to find an expression for a stabilizing controller, let us consider a candidate Lyapunov function: V = ( δq 4 ) 2 + δq T 3δq k 4 log 3 ki 2 i= (7) 3 (ki 2 ω2 i ) where, k, k 2 and k 3 are positive constant. We mention that the log-term in our Lyapunov function is motivated by the Lyapunov function introduced in Ref. [4]. Now, differentiating V with respect to time and making use of the fact that Ω(ω) is a skew-symmetric matrix leads to following expression for i= V : V = δq T 3ω + k 4 ω T k 2 ω2 k 2 2 ω2 2 k 2 3 ω2 3 ω (8) Now, substituting for ω from Eq. (b) in the above expression leads to V = δq T 3ω + k 4 ω T k 2 ω2 k 2 2 ω2 2 k 2 3 ω2 3 I ( ωiω + u) (9) Now, making use of the fact that ω is a skew-symmetric matrix, we get the following expression for V Now, if we choose control law to be V = δq T 3ω + k 4 ω T k 2 ω2 k 2 2 ω2 2 k 2 3 ω2 3 I u () u = IK ω (δq 3 + k 8 ω), K ω = k 4 k 2 ω2 k 4 k 2 2 ω2 2 k 4 k 2 3 ω2 3 () then V reduces to the following negative semi-definite function V = k 8 ω T ω (2) 4 of 2

5 It is clear that V is positive definite for the domain where ω has to lie within the hyper-rectangle with sides 2k i, i =, 2, 3. Thus, we can state that: ω i k i, i =, 2, 3 (3) Since V and V >, V is only negative semi-definite. However, we can easily show that δq, ω L. 5, 6 Further from the integral of Eq. (2), it follows that ω L L 2 and therefore from Barbalat s Lemma ω as t. Finally, using LaSalle s invariance principle 6 3, 5 we can show that δq as t. IV. Adaptation Law for Uncertain Inertia Matrix In this section, we seek to develop adaptation laws for inertia matrix I along with control law developed in the previous section to take care of any uncertainties in the inertia matrix. Let Î be the estimated value of the inertia matrix and I the inertia error matrix defined as follows: I = Î I (4) Further, we define a new variable J = I and analogous to I, we define: J = Ĵ J (5) Furthermore to enforce the symmetry constraint of J, we define a 6 vector: Θ = { J, J 2, J 3, J 22, J 23, J 33, } (6) Further, we make use of principle of equivalence 6 and assume that the expression for control vector of Eq. () is still valid. As a consequence of this, the applied control can be written as: u = ÎK ω (δq 3 + k 8 ω) (7) where, K ω = k 4 k 2 ω2 k 4 k 2 2 ω2 2 k 4 k 2 3 ω2 3 (8) 5 of 2

6 Now, let us consider a candidate Lyapunov function: V = ( δq 4 ) 2 + δq T 3δq k 4 log 3 ki 2 i= + 3 (ki 2 ω2 i ) i= 2 ΘT Γ Θ (9) Differentiating the above expression w.r.t. t and using the fact that ω is a skew-symmetric matrix leads to the following expression for V V = δq T 3ω + ω T K ω I u + Θ T Γ Θ (2) Now, substituting for u from Eq. (7) leads to [ ] V = δq T 3ω + ω T K ω I ÎK ω (δq 3 + k 8 ω) + Θ T Γ Θ (2a) Now, making use of Eq. (5), we have: ) I Î = JĴ = (Ĵ J Ĵ = JÎ, = (22a) Now, substitution of Eq. (22a) in Eq. (2a) leads to ) V = δq T 3ω ω T K ω ( JÎ K ω (δq 3 + k 8 ω) + Θ T Γ Θ = k 8 ω T ω + ω T K ω JÎK ω (δq 3 + k 8 ω) + Θ T Γ Θ (23a) Further, we can write: y y 2 y 3 JK ω ω = MΘ, M = y y 2 y 3, y = K ω ω (24) y y 2 y 3 Now, substitution of Eq. 24 in Eq. 23a leads to V = k 8 ω T ω + Θ T M T ÎK ω (δq 3 + k 8 ω) + Θ T Γ Θ (25a) 6 of 2

7 So, if we choose adaptation law to be Θ = ΓM T ÎK ω (δq 3 + k 8 ω) (26) Then, V reduces to the following negative semi-definite function: V = k 8 ω T ω (27) Since V and V >, V is only negative semi-definite. Once again, we can easily show that δq, ω, Θ L. 5, 6 Further from the integral of Eq. (27), it follows that ω L L 2 and therefore from Barbalat s Lemma ω as t. Finally, using LaSalle s invariance principle 6 3, 5 we can show that δq as t. Further, notice that it is easy to construct Ĵ = J from the expression for Θ and further, an expression for Î can be obtained by making use of the fact that ÎĴ = : ÎĴ + Î Ĵ = Î = Î ĴÎ (28a) (28b) Finally, notice that we can only guarantee that Θ L which means that estimated inertia matrix is bounded and there is no guarantee that estimated inertia matrix will converge to the true inertia matrix. Furthermore, one can invoke parameter projection 6 to guarantee that Ĵ is non-singular and positive-definite. V. Numerical Results The control laws presented in this paper are illustrated in this section for a particular attitude regulating maneuver. We consider following two simulation test cases with different initial conditions to demonstrate the effectiveness of the adaptive control laws. Test Case : q(t ) = {,,, } T, q(t f ) = {,.5, } T.75, ω(t ) = {,, } T, ω(t f ) = {,, } T Test Case 2: q(t ) = {,,, } T, q(t f ) = {,.5, } T.75, ω(t ) = {3, 3, 3} T, ω(t f ) = {,, } T 7 of 2

8 The angular velocity bounds for both the test cases are chosen to be: k = k 2 = k 3 = 4rad/sec. The true and initial inertia matrices of the rigid body are assumed to be: I = , Î = The various tuning parameters for the adaptive controllers for both the test cases are selected as follows: Γ = 2, k 8 =.5, k 4 = We mention that the feedback gain k 8 is deliberately chosen to be a small number so that inertia matrix term in the controller expression can dominate the pure feedback term. Fig. shows the various plots for the test case. Figs. (a) and (b) show the rigid body attitude (q) and angular velocity (ω), respectively. The corresponding commanded control input is shown in Fig. (c). The blue dashed line in these plot corresponds to control law without adapting for various inertia parameters while the red solid line in these plots corresponds to adaptive control law. From these plots it is clear that attitude error goes to zero over the time and rigid body angular velocity is well with in the prescribed bounds. Further, Fig. (d) shows the plots of estimated rigid body inertia parameters corresponding to update law of Eq. (26). From these plots, it is clear that the regulation performance of the controllers is marginally better with the adaptation of uncertain inertia matrix for test case. To show the effectiveness of the adaptation laws, let us consider the simulation results for test case 2 shown in Fig. 2. Figs. 2(a) and 2(b) show the rigid body attitude (q) and angular velocity (ω) plots for test case 2, respectively. The corresponding commanded control input is shown in Fig. 2(c). Once again, the blue dashed line in these plot corresponds to control law without adapting for various inertia parameters while the red solid line in these plots corresponds to adaptive control law. From these plots it is clear that although rigid body angular velocity is well with in the prescribed bounds with or without the adaptation of inertia matrix but attitude and angular velocity error converge to zero when the adaptation for uncertain inertia matrix is on. Further, Fig. (d) shows the plots of estimated rigid body inertia parameters corresponding to update law of Eq. (26). From these plots, it is clear that the regulation performance of the controllers is much better with the adaptation of uncertain inertia matrix for test case 2. It is worthwhile to notice 8 of 2

9 q.5 Adaptation off Adaptation on q 2 q 3 q 4 (a) q vs. ω 3 ω ω Adaptation off Adaptation on (b) ω vs. 2 u u u (c) u vs Δ I ij (d) I ij vs. Figure. Simulation Results for Test Case. 9 of 2

10 that the only difference between the two test cases is the initial conditions for the angular velocity vector ω. These results completely support the theoretical result that performance of the controller can be improved with the adaptation of unknown inertia matrix. VI. Concluding Remarks An asymptotically stable adaptive controller has been designed for rigid body attitude control which explicitly takes into consideration the bounds on angular velocity. The adaptive control formulations in this paper is based upon Lyapunov s direct stability theorem and imposes the exact kinematic equations at the velocity level while taking care of model uncertainties at the acceleration level. The proposed control law is shown to work well in the presence of bounded angular velocity constraints fully consistent with the asymptotic stability analysis presented. While, the simulation results presented in this paper merely illustrate formulations for a particular attitude maneuver, further testing would be required to reach any conclusions about the efficacy of the control and adaptation laws for tracking arbitrary maneuvers. References Schaub, H. and Junkins, J. L., Analytical Mechanics of Space Systems, AIAA Education Series, AIAA, Junkins, J. L. and Turner, J. D., Optimal Spacecraft Rotational Maneuvers, Elsevier Science Publishers, Shuster, M. D., A Survey of Attitude Representations, Journal of the Astronautical Sciences, Vol. 4, No. 4, October December 993, pp Junkins, J. L. and Singla, P., How Nonlinear Is It? A Tutorial on Nonlinearity of Orbit and Attitude Dynamics, Journal of Astronautical Sciences, Vol. 52, No. -2, 24, pp. 7 6, keynote paper. 5 Schaub, H., Akella, M., and Junkins, J. L., Adaptive Control of Nonlinear Attitude Motions Realizing Linear Closed Loop Dynamics, AIAA Journal of Guidance, Control, and Dynamics, Vol. 24, No., 2, pp Carrington, C. K. and Junkins, J. L., Optimal nonlinear feedback control for spacecraft attitude maneuvers, AIAA Journal of Guidance, Control, and Dynamics, Vol. 9, No., 986, pp Tsiotras, P., Further Passivity Results for the Attitude Control Problem, IEEE Transactions on Automatic Contorl, Vol. 43, No., Nov 998, pp Krstic, M. and Tsiotras, P., Inverse optimality results for the attitude motion of a rigid spacecraft, American Control Conference, 997. Proceedings of the 997, Vol. 3, 4-6 Jun 997, pp vol.3. 9 Tsiotras, P., Stabilization and optimality results for the attitude control problem, AIAA Journal of Guidance, Control, and Dynamics, Vol. 9, No. 4, 996, pp Tsiotras, P., A passivity approach to attitude stabilization using nonredundant kinematic parameterizations, Decision and Control, 995., Proceedings of the 34th IEEE Conference on, Vol., 3-5 Dec 995, pp vol.. Junkins, J. L., Akella, M. R., and Robinett, R. D., Nonlinear Adaptive Control of Spacecraft Maneuvers, AIAA Journal of Guidance, Control, and Dynamics, Vol. 2, No. 6, 997, pp Wen, J. T. Y. and Kreutz-Delgado, K., The attitude control problem, Automatic Control, IEEE Transactions on, Vol. 36, No., Oct 99, pp of 2

11 q q 2 q 3 q 4 Adaptation off Adaptation on (a) q vs. ω ω 2 ω Adaptation off Adaptation on (b) ω vs. u 5 Adaptation off Adaptation on u u (c) u vs Δ I ij (d) I ij vs. Figure 2. Simulation Results for Test Case 2. of 2

12 3 Tanygin, S., Generalization of Adaptive Attitude Tracking, AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Monterey, CA, USA, August Ngo, K., Mahony, R., and Jiang, Z.-P., Integrator backstepping design for motion systems with velocity constraint, Control Conference, 24. 5th Asian, Vol., July 24, pp Vol.. 5 Sastry, S., Nonlinear Systems: Analysis, Stability and Control, Springer-Verlag, NY, USA, Ionnaou, P. A. and Sun, J., Robust Adaptive Control, Prentice Hall Inc., NJ, of 2

Principal Rotation Representations of Proper NxN Orthogonal Matrices

Principal Rotation Representations of Proper NxN Orthogonal Matrices Principal Rotation Representations of Proper NxN Orthogonal Matrices Hanspeter Schaub Panagiotis siotras John L. Junkins Abstract hree and four parameter representations of x orthogonal matrices are extended

More information

Spacecraft Dynamics and Control. An Introduction

Spacecraft 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 information

Design-Simulation-Optimization Package for a Generic 6-DOF Manipulator with a Spherical Wrist

Design-Simulation-Optimization Package for a Generic 6-DOF Manipulator with a Spherical Wrist Design-Simulation-Optimization Package for a Generic 6-DOF Manipulator with a Spherical Wrist MHER GRIGORIAN, TAREK SOBH Department of Computer Science and Engineering, U. of Bridgeport, USA ABSTRACT Robot

More information

CONTRIBUTIONS TO THE AUTOMATIC CONTROL OF AERIAL VEHICLES

CONTRIBUTIONS 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 I3S-CNRS Sophia Antipolis, CONDOR team Project ANR SCUAV Supervisors: Pascal MORIN,

More information

Dynamics. Basilio Bona. DAUIN-Politecnico di Torino. Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 1 / 30

Dynamics. Basilio Bona. DAUIN-Politecnico di Torino. Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 1 / 30 Dynamics Basilio Bona DAUIN-Politecnico di Torino 2009 Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 1 / 30 Dynamics - Introduction In order to determine the dynamics of a manipulator, it is

More information

Operational Space Control for A Scara Robot

Operational Space Control for A Scara Robot Operational Space Control for A Scara Robot Francisco Franco Obando D., Pablo Eduardo Caicedo R., Oscar Andrés Vivas A. Universidad del Cauca, {fobando, pacaicedo, avivas }@unicauca.edu.co Abstract This

More information

MECH 5105 Orbital Mechanics and Control. Steve Ulrich Carleton University Ottawa, ON, Canada

MECH 5105 Orbital Mechanics and Control. Steve Ulrich Carleton University Ottawa, ON, Canada MECH 5105 Orbital Mechanics and Control Steve Ulrich Carleton University Ottawa, ON, Canada 2 Copyright c 2015 by Steve Ulrich 3 4 Course Outline About the Author Steve Ulrich is an Assistant Professor

More information

INSTRUCTOR WORKBOOK Quanser Robotics Package for Education for MATLAB /Simulink Users

INSTRUCTOR WORKBOOK Quanser Robotics Package for Education for MATLAB /Simulink Users INSTRUCTOR WORKBOOK for MATLAB /Simulink Users Developed by: Amir Haddadi, Ph.D., Quanser Peter Martin, M.A.SC., Quanser Quanser educational solutions are powered by: CAPTIVATE. MOTIVATE. GRADUATE. PREFACE

More information

2. Dynamics, Control and Trajectory Following

2. Dynamics, Control and Trajectory Following 2. Dynamics, Control and Trajectory Following This module Flying vehicles: how do they work? Quick refresher on aircraft dynamics with reference to the magical flying space potato How I learned to stop

More information

AMIT K. SANYAL. 2001-2004 Ph.D. in Aerospace Engineering, University of Michigan, Ann Arbor, MI. Date of completion:

AMIT K. SANYAL. 2001-2004 Ph.D. in Aerospace Engineering, University of Michigan, Ann Arbor, MI. Date of completion: AMIT K. SANYAL Office Home 305 Holmes Hall 3029 Lowrey Avenue Mechanical Engineering Apartment # N-2211 University of Hawaii at Manoa Honolulu, HI 96822 Honolulu, HI 96822 480-603-8938 808-956-2142 aksanyal@hawaii.edu

More information

Stability Analysis of Small Satellite Formation Flying and Reconfiguration Missions in Deep Space

Stability Analysis of Small Satellite Formation Flying and Reconfiguration Missions in Deep Space Stability Analysis of Small Satellite Formation Flying and Reconfiguration Missions in Deep Space Saptarshi Bandyopadhyay, Chakravarthini M. Saaj, and Bijnan Bandyopadhyay, Member, IEEE Abstract Close-proximity

More information

Linear-Quadratic Optimal Controller 10.3 Optimal Linear Control Systems

Linear-Quadratic Optimal Controller 10.3 Optimal Linear Control Systems Linear-Quadratic Optimal Controller 10.3 Optimal Linear Control Systems In Chapters 8 and 9 of this book we have designed dynamic controllers such that the closed-loop systems display the desired transient

More information

Research Article End-Effector Trajectory Tracking Control of Space Robot with L 2 Gain Performance

Research Article End-Effector Trajectory Tracking Control of Space Robot with L 2 Gain Performance Mathematical Problems in Engineering Volume 5, Article ID 7534, 9 pages http://dx.doi.org/.55/5/7534 Research Article End-Effector Trajectory Tracking Control of Space Robot with L Gain Performance Haibo

More information

Force/position control of a robotic system for transcranial magnetic stimulation

Force/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 information

Bead moving along a thin, rigid, wire.

Bead moving along a thin, rigid, wire. Bead moving along a thin, rigid, wire. odolfo. osales, Department of Mathematics, Massachusetts Inst. of Technology, Cambridge, Massachusetts, MA 02139 October 17, 2004 Abstract An equation describing

More information

Optimal Design of α-β-(γ) Filters

Optimal Design of α-β-(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 information

Stabilizing a Gimbal Platform using Self-Tuning Fuzzy PID Controller

Stabilizing a Gimbal Platform using Self-Tuning Fuzzy PID Controller Stabilizing a Gimbal Platform using Self-Tuning Fuzzy PID Controller Nourallah Ghaeminezhad Collage Of Automation Engineering Nuaa Nanjing China Wang Daobo Collage Of Automation Engineering Nuaa Nanjing

More information

4 Lyapunov Stability Theory

4 Lyapunov Stability Theory 4 Lyapunov Stability Theory In this section we review the tools of Lyapunov stability theory. These tools will be used in the next section to analyze the stability properties of a robot controller. We

More information

1 KINEMATICS OF MOVING FRAMES

1 KINEMATICS OF MOVING FRAMES 1 1 KINEMATICS OF MOVING FRAMES 1.1 Rotation of Reference Frames We denote through a subscript the specific reference system of a vector. Let a vector expressed in the inertial frame be denoted as γx,

More information

Dimension Theory for Ordinary Differential Equations

Dimension Theory for Ordinary Differential Equations Vladimir A. Boichenko, Gennadij A. Leonov, Volker Reitmann Dimension Theory for Ordinary Differential Equations Teubner Contents Singular values, exterior calculus and Lozinskii-norms 15 1 Singular values

More information

Rotation: Moment of Inertia and Torque

Rotation: 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 information

Optimal Reconfiguration of Formation Flying Satellites

Optimal Reconfiguration of Formation Flying Satellites Proceedings of the th IEEE Conference on Decision and Control, and the European Control Conference 5 Seville, Spain, December -5, 5 MoA.6 Optimal Reconfiguration of Formation Flying Satellites Oliver Junge

More information

A PAIR OF MEASURES OF ROTATIONAL ERROR FOR AXISYMMETRIC ROBOT END-EFFECTORS

A PAIR OF MEASURES OF ROTATIONAL ERROR FOR AXISYMMETRIC ROBOT END-EFFECTORS A PAIR OF MEASURES OF ROTATIONAL ERROR FOR AXISYMMETRIC ROBOT END-EFFECTORS Sébastien Briot, Ilian A. Bonev Department of Automated Manufacturing Engineering École de technologie supérieure (ÉTS), Montreal,

More information

Example 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x

Example 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 information

Physics in the Laundromat

Physics in the Laundromat Physics in the Laundromat Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (Aug. 5, 1997) Abstract The spin cycle of a washing machine involves motion that is stabilized

More information

Problem of the gyroscopic stabilizer damping

Problem of the gyroscopic stabilizer damping Applied and Computational Mechanics 3 (2009) 205 212 Problem of the gyroscopic stabilizer damping J. Šklíba a, a Faculty of Mechanical Engineering, Technical University in Liberec, Studentská 2, 461 17,

More information

Simulation of Trajectories and Comparison of Joint Variables for Robotic Manipulator Using Multibody Dynamics (MBD)

Simulation of Trajectories and Comparison of Joint Variables for Robotic Manipulator Using Multibody Dynamics (MBD) Simulation of Trajectories and Comparison of Joint Variables for Robotic Manipulator Using Multibody Dynamics (MBD) Jatin Dave Assistant Professor Nirma University Mechanical Engineering Department, Institute

More information

Véronique PERDEREAU ISIR UPMC 6 mars 2013

Véronique PERDEREAU ISIR UPMC 6 mars 2013 Véronique PERDEREAU ISIR UPMC mars 2013 Conventional methods applied to rehabilitation robotics Véronique Perdereau 2 Reference Robot force control by Bruno Siciliano & Luigi Villani Kluwer Academic Publishers

More information

geometric transforms

geometric transforms geometric transforms 1 linear algebra review 2 matrices matrix and vector notation use column for vectors m 11 =[ ] M = [ m ij ] m 21 m 12 m 22 =[ ] v v 1 v = [ ] T v 1 v 2 2 3 matrix operations addition

More information

Metrics on SO(3) and Inverse Kinematics

Metrics on SO(3) and Inverse Kinematics Mathematical Foundations of Computer Graphics and Vision Metrics on SO(3) and Inverse Kinematics Luca Ballan Institute of Visual Computing Optimization on Manifolds Descent approach d is a ascent direction

More information

Precise 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 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 E-mail : renuka_mee@nitc.ac.in,

More information

1 Scalars, Vectors and Tensors

1 Scalars, Vectors and Tensors DEPARTMENT OF PHYSICS INDIAN INSTITUTE OF TECHNOLOGY, MADRAS PH350 Classical Physics Handout 1 8.8.2009 1 Scalars, Vectors and Tensors In physics, we are interested in obtaining laws (in the form of mathematical

More information

A Control Scheme for Industrial Robots Using Artificial Neural Networks

A Control Scheme for Industrial Robots Using Artificial Neural Networks A Control Scheme for Industrial Robots Using Artificial Neural Networks M. Dinary, Abou-Hashema M. El-Sayed, Abdel Badie Sharkawy, and G. Abouelmagd unknown dynamical plant is investigated. A layered neural

More information

Least-Squares Intersection of Lines

Least-Squares Intersection of Lines Least-Squares Intersection of Lines Johannes Traa - UIUC 2013 This write-up derives the least-squares solution for the intersection of lines. In the general case, a set of lines will not intersect at a

More information

Origins of the Unusual Space Shuttle Quaternion Definition

Origins of the Unusual Space Shuttle Quaternion Definition 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition 5-8 January 2009, Orlando, Florida AIAA 2009-43 Origins of the Unusual Space Shuttle Quaternion Definition

More information

Motion Control of 3 Degree-of-Freedom Direct-Drive Robot. Rutchanee Gullayanon

Motion Control of 3 Degree-of-Freedom Direct-Drive Robot. Rutchanee Gullayanon Motion Control of 3 Degree-of-Freedom Direct-Drive Robot A Thesis Presented to The Academic Faculty by Rutchanee Gullayanon In Partial Fulfillment of the Requirements for the Degree Master of Engineering

More information

Online Tuning of Artificial Neural Networks for Induction Motor Control

Online Tuning of Artificial Neural Networks for Induction Motor Control Online Tuning of Artificial Neural Networks for Induction Motor Control A THESIS Submitted by RAMA KRISHNA MAYIRI (M060156EE) In partial fulfillment of the requirements for the award of the Degree of MASTER

More information

Lecture L22-2D Rigid Body Dynamics: Work and Energy

Lecture L22-2D Rigid Body Dynamics: Work and Energy J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for

More information

A Passivity Measure Of Systems In Cascade Based On Passivity Indices

A Passivity Measure Of Systems In Cascade Based On Passivity Indices 49th IEEE Conference on Decision and Control December 5-7, Hilton Atlanta Hotel, Atlanta, GA, USA A Passivity Measure Of Systems In Cascade Based On Passivity Indices Han Yu and Panos J Antsaklis Abstract

More information

On Motion of Robot End-Effector using the Curvature Theory of Timelike Ruled Surfaces with Timelike Directrix

On Motion of Robot End-Effector using the Curvature Theory of Timelike Ruled Surfaces with Timelike Directrix Malaysian Journal of Mathematical Sciences 8(2): 89-204 (204) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: http://einspem.upm.edu.my/journal On Motion of Robot End-Effector using the Curvature

More information

Geometric Adaptive Control of Quadrotor UAVs Transporting a Cable-Suspended Rigid Body

Geometric Adaptive Control of Quadrotor UAVs Transporting a Cable-Suspended Rigid Body Geometric Adaptive Control of Quadrotor UAVs Transporting a Cable-Suspended Rigid Body Taeyoung Lee Abstract This paper is focused on tracking control for a rigid body payload that is connected to an arbitrary

More information

AAS/AIAA Astrodynamics Specialists Conference

AAS/AIAA Astrodynamics Specialists Conference Paper AAS 03-548 RELATING POSITION UNCERTAINTY TO MAXIMUM CONJUNCTION PROBABILITY Salvatore Alfano Copyright 2003 by The Aerospace Corporation. Published by the American Astronautical Society, with permission

More information

System Modeling and Control for Mechanical Engineers

System Modeling and Control for Mechanical Engineers Session 1655 System Modeling and Control for Mechanical Engineers Hugh Jack, Associate Professor Padnos School of Engineering Grand Valley State University Grand Rapids, MI email: jackh@gvsu.edu Abstract

More information

Lecture L19 - Vibration, Normal Modes, Natural Frequencies, Instability

Lecture L19 - Vibration, Normal Modes, Natural Frequencies, Instability S. Widnall 16.07 Dynamics Fall 2009 Version 1.0 Lecture L19 - Vibration, Normal Modes, Natural Frequencies, Instability Vibration, Instability An important class of problems in dynamics concerns the free

More information

Attitude Control and Dynamics of Solar Sails

Attitude 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 information

Introduction to Engineering System Dynamics

Introduction 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 information

ACTUATOR DESIGN FOR ARC WELDING ROBOT

ACTUATOR DESIGN FOR ARC WELDING ROBOT ACTUATOR DESIGN FOR ARC WELDING ROBOT 1 Anurag Verma, 2 M. M. Gor* 1 G.H Patel College of Engineering & Technology, V.V.Nagar-388120, Gujarat, India 2 Parul Institute of Engineering & Technology, Limda-391760,

More information

Lecture L18 - Exploring the Neighborhood: the Restricted Three-Body Problem

Lecture L18 - Exploring the Neighborhood: the Restricted Three-Body Problem S. Widnall 16.07 Dynamics Fall 008 Version 1.0 Lecture L18 - Exploring the Neighborhood: the Restricted Three-Body Problem The Three-Body Problem In Lecture 15-17, we presented the solution to the two-body

More information

A New Nature-inspired Algorithm for Load Balancing

A New Nature-inspired Algorithm for Load Balancing A New Nature-inspired Algorithm for Load Balancing Xiang Feng East China University of Science and Technology Shanghai, China 200237 Email: xfeng{@ecusteducn, @cshkuhk} Francis CM Lau The University of

More information

THE UNIVERSITY OF TEXAS AT AUSTIN Department of Aerospace Engineering and Engineering Mechanics. EM 311M - DYNAMICS Spring 2012 SYLLABUS

THE UNIVERSITY OF TEXAS AT AUSTIN Department of Aerospace Engineering and Engineering Mechanics. EM 311M - DYNAMICS Spring 2012 SYLLABUS THE UNIVERSITY OF TEXAS AT AUSTIN Department of Aerospace Engineering and Engineering Mechanics EM 311M - DYNAMICS Spring 2012 SYLLABUS UNIQUE NUMBERS: 13815, 13820, 13825, 13830 INSTRUCTOR: TIME: Dr.

More information

Control and Navigation Framework for Quadrotor Helicopters

Control and Navigation Framework for Quadrotor Helicopters Control and Navigation Framework for Quadrotor Helicopters Amr Nagaty, Sajad Saeedi, Carl Thibault, Mae Seto and Howard Li Abstract This paper presents the development of a nonlinear quadrotor simulation

More information

Basic 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 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 information

Advantages of Auto-tuning for Servo-motors

Advantages of Auto-tuning for Servo-motors Advantages of for Servo-motors 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 information

CIS 536/636 Introduction to Computer Graphics. Kansas State University. CIS 536/636 Introduction to Computer Graphics

CIS 536/636 Introduction to Computer Graphics. Kansas State University. CIS 536/636 Introduction to Computer Graphics 2 Lecture Outline Animation 2 of 3: Rotations, Quaternions Dynamics & Kinematics William H. Hsu Department of Computing and Information Sciences, KSU KSOL course pages: http://bit.ly/hgvxlh / http://bit.ly/evizre

More information

General model of a structure-borne sound source and its application to shock vibration

General model of a structure-borne sound source and its application to shock vibration General model of a structure-borne sound source and its application to shock vibration Y. Bobrovnitskii and T. Tomilina Mechanical Engineering Research Institute, 4, M. Kharitonievky Str., 101990 Moscow,

More information

Computer Animation. Lecture 2. Basics of Character Animation

Computer Animation. Lecture 2. Basics of Character Animation Computer Animation Lecture 2. Basics of Character Animation Taku Komura Overview Character Animation Posture representation Hierarchical structure of the body Joint types Translational, hinge, universal,

More information

Stability Of Structures: Basic Concepts

Stability Of Structures: Basic Concepts 23 Stability Of Structures: Basic Concepts ASEN 3112 Lecture 23 Slide 1 Objective This Lecture (1) presents basic concepts & terminology on structural stability (2) describes conceptual procedures for

More information

Chapter 24 Physical Pendulum

Chapter 24 Physical Pendulum Chapter 4 Physical Pendulum 4.1 Introduction... 1 4.1.1 Simple Pendulum: Torque Approach... 1 4. Physical Pendulum... 4.3 Worked Examples... 4 Example 4.1 Oscillating Rod... 4 Example 4.3 Torsional Oscillator...

More information

State of Stress at Point

State of Stress at Point State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

More information

Intelligent Mechatronic Model Reference Theory for Robot Endeffector

Intelligent Mechatronic Model Reference Theory for Robot Endeffector , pp.165-172 http://dx.doi.org/10.14257/ijunesst.2015.8.1.15 Intelligent Mechatronic Model Reference Theory for Robot Endeffector Control Mohammad sadegh Dahideh, Mohammad Najafi, AliReza Zarei, Yaser

More information

Worldwide, space agencies are increasingly exploiting multi-body dynamical structures for their most

Worldwide, space agencies are increasingly exploiting multi-body dynamical structures for their most Coupled Orbit-Attitude Dynamics in the Three-Body Problem: a Family of Orbit-Attitude Periodic Solutions Davide Guzzetti and Kathleen C. Howell Purdue University, Armstrong Hall of Engineering, 71 W. Stadium

More information

A Direct Numerical Method for Observability Analysis

A Direct Numerical Method for Observability Analysis IEEE TRANSACTIONS ON POWER SYSTEMS, VOL 15, NO 2, MAY 2000 625 A Direct Numerical Method for Observability Analysis Bei Gou and Ali Abur, Senior Member, IEEE Abstract This paper presents an algebraic method

More information

Human-like Arm Motion Generation for Humanoid Robots Using Motion Capture Database

Human-like Arm Motion Generation for Humanoid Robots Using Motion Capture Database Human-like Arm Motion Generation for Humanoid Robots Using Motion Capture Database Seungsu Kim, ChangHwan Kim and Jong Hyeon Park School of Mechanical Engineering Hanyang University, Seoul, 133-791, Korea.

More information

Automatic Synthesis of a Planar Linkage Mechanism

Automatic Synthesis of a Planar Linkage Mechanism Automatic Synthesis of a Planar Linkage Mechanism Yoon Young Kim Seoul National University 2 Our Goal: Automatic Mechanism Synthesis?? 3 Research Motivation Arrow Model at Brandeis Univ. Space Robot at

More information

3D Tranformations. CS 4620 Lecture 6. Cornell CS4620 Fall 2013 Lecture 6. 2013 Steve Marschner (with previous instructors James/Bala)

3D Tranformations. CS 4620 Lecture 6. Cornell CS4620 Fall 2013 Lecture 6. 2013 Steve Marschner (with previous instructors James/Bala) 3D Tranformations CS 4620 Lecture 6 1 Translation 2 Translation 2 Translation 2 Translation 2 Scaling 3 Scaling 3 Scaling 3 Scaling 3 Rotation about z axis 4 Rotation about z axis 4 Rotation about x axis

More information

PID, LQR and LQR-PID on a Quadcopter Platform

PID, LQR and LQR-PID on a Quadcopter Platform PID, LQR and LQR-PID on a Quadcopter Platform Lucas M. Argentim unielargentim@fei.edu.br Willian C. Rezende uniewrezende@fei.edu.br Paulo E. Santos psantos@fei.edu.br Renato A. Aguiar preaguiar@fei.edu.br

More information

Robot Task-Level Programming Language and Simulation

Robot Task-Level Programming Language and Simulation Robot Task-Level Programming Language and Simulation M. Samaka Abstract This paper presents the development of a software application for Off-line robot task programming and simulation. Such application

More information

Kinematical Animation. lionel.reveret@inria.fr 2013-14

Kinematical Animation. lionel.reveret@inria.fr 2013-14 Kinematical Animation 2013-14 3D animation in CG Goal : capture visual attention Motion of characters Believable Expressive Realism? Controllability Limits of purely physical simulation : - little interactivity

More information

Kinematic Optimal Design on a New Robotic Platform for Stair Climbing

Kinematic Optimal Design on a New Robotic Platform for Stair Climbing Kinematic Optimal Design on a New Robotic Platform for Stair Climbing Byung Hoon Seo, Hyun Gyu Kim, and Tae Won Seo Abstract Stair climbing is one of critical issues for field robots to widen applicable

More information

Numerical Solution of Differential Equations

Numerical Solution of Differential Equations Numerical Solution of Differential Equations Dr. Alvaro Islas Applications of Calculus I Spring 2008 We live in a world in constant change We live in a world in constant change We live in a world in constant

More information

Orbits of the Lennard-Jones Potential

Orbits of the Lennard-Jones Potential Orbits of the Lennard-Jones Potential Prashanth S. Venkataram July 28, 2012 1 Introduction The Lennard-Jones potential describes weak interactions between neutral atoms and molecules. Unlike the potentials

More information

CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation

CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in

More information

On-line trajectory planning of robot manipulator s end effector in Cartesian Space using quaternions

On-line trajectory planning of robot manipulator s end effector in Cartesian Space using quaternions On-line trajectory planning of robot manipulator s end effector in Cartesian Space using quaternions Ignacio Herrera Aguilar and Daniel Sidobre (iherrera, daniel)@laas.fr LAAS-CNRS Université Paul Sabatier

More information

dspace DSP DS-1104 based State Observer Design for Position Control of DC Servo Motor

dspace DSP DS-1104 based State Observer Design for Position Control of DC Servo Motor dspace DSP DS-1104 based State Observer Design for Position Control of DC Servo Motor Jaswandi Sawant, Divyesh Ginoya Department of Instrumentation and control, College of Engineering, Pune. ABSTRACT This

More information

Robust Design of Inertial Measurement Units Based on Accelerometers

Robust Design of Inertial Measurement Units Based on Accelerometers Robust Design of Inertial Measurement Units Based on Accelerometers Zhongkai Qin, Luc Baron and Lionel Birglen {zhongkai.qin, luc.baron, lionel.birglen}@polymtl.ca Department of Mechanical Engineering,

More information

Kinematics and Dynamics of Mechatronic Systems. Wojciech Lisowski. 1 An Introduction

Kinematics and Dynamics of Mechatronic Systems. Wojciech Lisowski. 1 An Introduction Katedra Robotyki i Mechatroniki Akademia Górniczo-Hutnicza w Krakowie Kinematics and Dynamics of Mechatronic Systems Wojciech Lisowski 1 An Introduction KADOMS KRIM, WIMIR, AGH Kraków 1 The course contents:

More information

Adaptive Control for Robot Manipulators Under Ellipsoidal Task Space Constraints

Adaptive Control for Robot Manipulators Under Ellipsoidal Task Space Constraints 22 IEEE/RSJ International Conference on Intelligent Robots and Systems October 7-2, 22. Vilamoura, Algarve, Portugal Adaptive Control for Robot Manipulators Under Ellipsoidal Task Space Constraints Keng

More information

Unit 21 Influence Coefficients

Unit 21 Influence Coefficients Unit 21 Influence Coefficients Readings: Rivello 6.6, 6.13 (again), 10.5 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Have considered the vibrational behavior of

More information

Strategies for Continuous Mars Habitation with a Limited Number of Cycler Vehicles

Strategies for Continuous Mars Habitation with a Limited Number of Cycler Vehicles Strategies for Continuous Mars Habitation with a Limited Number of Cycler Vehicles Damon Landau James M. Longuski October 2005 Prepared for Dr. Buzz Aldrin Starcraft Enterprises By Purdue University School

More information

A MONTE CARLO DISPERSION ANALYSIS OF A ROCKET FLIGHT SIMULATION SOFTWARE

A MONTE CARLO DISPERSION ANALYSIS OF A ROCKET FLIGHT SIMULATION SOFTWARE A MONTE CARLO DISPERSION ANALYSIS OF A ROCKET FLIGHT SIMULATION SOFTWARE F. SAGHAFI, M. KHALILIDELSHAD Department of Aerospace Engineering Sharif University of Technology E-mail: saghafi@sharif.edu Tel/Fax:

More information

Effect of Remote Back-Up Protection System Failure on the Optimum Routine Test Time Interval of Power System Protection

Effect of Remote Back-Up Protection System Failure on the Optimum Routine Test Time Interval of Power System Protection Effect of Remote Back-Up Protection System Failure on the Optimum Routine Test Time Interval of Power System Protection Y. Damchi* and J. Sadeh* (C.A.) Abstract: Appropriate operation of protection system

More information

Effect of Remote Back-Up Protection System Failure on the Optimum Routine Test Time Interval of Power System Protection

Effect of Remote Back-Up Protection System Failure on the Optimum Routine Test Time Interval of Power System Protection Effect of Remote Back-Up Protection System Failure on the Optimum Routine Test Time Interval of Power System Protection Y. Damchi* and J. Sadeh* (C.A.) Abstract: Appropriate operation of protection system

More information

1 Differential Drive Kinematics

1 Differential Drive Kinematics CS W4733 NOTES - Differential Drive Robots Note: these notes were compiled from Dudek and Jenkin, Computational Principles of Mobile Robotics. 1 Differential Drive Kinematics Many mobile robots use a drive

More information

Particle Swarm and Differential Evolution Optimization for stochastic inversion of post-stack seismic data

Particle Swarm and Differential Evolution Optimization for stochastic inversion of post-stack seismic data P-207 Particle Swarm and Differential Evolution Optimization for stochastic inversion of post-stack seismic data Puneet Saraswat *, Indian School of Mines, Dhanbad, Dr. Ravi Prakash Srivastava, Scientist,

More information

animation animation shape specification as a function of time

animation animation shape specification as a function of time animation animation shape specification as a function of time animation representation many ways to represent changes with time intent artistic motion physically-plausible motion efficiency control typically

More information

Time Domain and Frequency Domain Techniques For Multi Shaker Time Waveform Replication

Time 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) 216-8440 This paper

More information

An Introduction to the Mofied Nodal Analysis

An Introduction to the Mofied Nodal Analysis An Introduction to the Mofied Nodal Analysis Michael Hanke May 30, 2006 1 Introduction Gilbert Strang provides an introduction to the analysis of electrical circuits in his book Introduction to Applied

More information

A Robust Estimator for Almost Global Attitude Feedback Tracking

A Robust Estimator for Almost Global Attitude Feedback Tracking A Robust Estimator for Almost Global Attitude Feedback Tracking Amit K. Sanyal Nikolaj Nordkvist This article presents a robust and almost global feedback attitude tracking control scheme in conjunction

More information

Robot Dynamics and Control

Robot Dynamics and Control Chapter 4 Robot Dynamics and Control This chapter presents an introduction to the dynamics and control of robot manipulators. We derive the equations of motion for a general open-chain manipulator and,

More information

Dynamics and Control of an Elastic Dumbbell Spacecraft in a Central Gravitational Field

Dynamics and Control of an Elastic Dumbbell Spacecraft in a Central Gravitational Field Dynamics Control of an Elastic Dumbbell Spacecraft in a Central Gravitational Field Amit K. Sanyal, Jinglai Shen, N. Harris McClamroch 1 Department of Aerospace Engineering University of Michigan Ann Arbor,

More information

Pre-requisites 2012-2013

Pre-requisites 2012-2013 Pre-requisites 2012-2013 Engineering Computation The student should be familiar with basic tools in Mathematics and Physics as learned at the High School level and in the first year of Engineering Schools.

More information

Geometric Constraints

Geometric Constraints Simulation in Computer Graphics Geometric Constraints Matthias Teschner Computer Science Department University of Freiburg Outline introduction penalty method Lagrange multipliers local constraints University

More information

DETERMINANTS. b 2. x 2

DETERMINANTS. b 2. x 2 DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

More information

Adaptive Control Using Combined Online and Background Learning Neural Network

Adaptive Control Using Combined Online and Background Learning Neural Network Adaptive Control Using Combined Online and Background Learning Neural Network Eric N. Johnson and Seung-Min Oh Abstract A new adaptive neural network (NN control concept is proposed with proof of stability

More information

Constraint satisfaction and global optimization in robotics

Constraint satisfaction and global optimization in robotics Constraint satisfaction and global optimization in robotics Arnold Neumaier Universität Wien and Jean-Pierre Merlet INRIA Sophia Antipolis 1 The design, validation, and use of robots poses a number of

More information

Force 7. Force Control

Force 7. Force Control 161 Force 7. Force Control Luigi Villani, Joris De Schutter A fundamental requirement for the success of a manipulation task is the capability to handle the physical contact between a robot and the environment.

More information

SAMPLE CHAPTERS UNESCO EOLSS PID CONTROL. Araki M. Kyoto University, Japan

SAMPLE CHAPTERS UNESCO EOLSS PID CONTROL. Araki M. Kyoto University, Japan PID CONTROL Araki M. Kyoto University, Japan Keywords: feedback control, proportional, integral, derivative, reaction curve, process with self-regulation, integrating process, process model, steady-state

More information

Transmission through the quadrupole mass spectrometer mass filter: The effect of aperture and harmonics

Transmission through the quadrupole mass spectrometer mass filter: The effect of aperture and harmonics Transmission through the quadrupole mass spectrometer mass filter: The effect of aperture and harmonics A. C. C. Voo, R. Ng, a) J. J. Tunstall, b) and S. Taylor Department of Electrical and Electronic

More information

The Matrix Elements of a 3 3 Orthogonal Matrix Revisited

The Matrix Elements of a 3 3 Orthogonal Matrix Revisited Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, Three-Dimensional Proper and Improper Rotation Matrices, I provided a derivation

More information

Indeterminate Analysis Force Method 1

Indeterminate Analysis Force Method 1 Indeterminate Analysis Force Method 1 The force (flexibility) method expresses the relationships between displacements and forces that exist in a structure. Primary objective of the force method is to

More information