摘要
任务空间是机械系统的一种典型位形空间,在几何上对应了李群SE(3)。任务空间的控制问题涵盖了丰富的研究对象,但由于任务空间复杂的空间结构,任务空间控制的相关理论研究进展缓慢,而且与工程应用相距甚远。本论文的研究是针对任务空间控制理论研究的缺陷和应用的需求开展的,着重关注任务空间控制的两个难点问题:由李群的矩阵表示带来的运算负担问题,以及无侧移运动控制问题。
论文的前半部分面向全任务空间控制问题,致力于减轻控制算法的运算负担。针对姿态控制和位姿控制问题,用四元数代数工具代替矩阵作为空间表示的手段,寻找可以简化计算、同时可以充分利用空间结构特点的控制方法,主要取得了以下三个方面的成果:
1.建立了单位四元数和归一化对偶四元数的李群结构,借助单位四元数李群Q_u和对偶四元数李群DQ_u的李代数,分别构造了姿态控制和位姿控制的广义比例—微分控制律,妥善处理了四元数表示方式带来的双平衡点问题,针对全驱动模型实现了镇定控制和跟踪控制。
2.使用单位四元数进行了最优姿态控制和旋转轨迹设计的研究:针对自由漂浮空间机器人卫星基座姿态自调整问题,借助Q_u的李代数,使用线性系统最优控制理论实现了具有能量优化特性的姿态控制;针对具有最小加速度约束的两点旋转插值问题,使用多项式样条给出了两点距离较小时的近似解析解。
3.使用对偶四元数位姿控制的思想,开展了多体系统控制问题的研究:针对多自由度机械臂,给出了具体模型与广义PD控制律相结合的设计实例;以多自由度机械臂为例推导了全任务空间镇定控制律;提出了双任务空间控制的概念,设计了自由漂浮空间机器人的基座最小扰动控制律。
另一方面,考察以飞行器为典型代表的无侧移运动,其模型特性导致难以进行统一的位姿控制,为此对质点轨迹规划和无侧移运动模型分析两个子问题展开研究。此外,作为无侧移运动的一种简化形式,地面移动机器人的控制需要寻找仅依赖角度测量信息的控制律。论文的后半部分着重研究无侧移运动控制的相关问题,取得了三个方面的成果:
4.实时轨迹规划:针对恒速质点运动这类典型运动形式,结合飞行器制导设计的思想,依靠视线方位角的变化信息产生法向加速度,从而完成实时轨迹规划。其中视线方位的描述可以使用李群或者旋量进行,由此分别得到了两种实时轨迹规划的方法,在作为制导律使用时,能够克服传统方法依赖通道解耦进行三维制导设计的缺陷,且实现了多约束制导的部分要求。
5.模型分析:对无侧移运动的微分平坦分析指出,运动维度的变化影响运动性能。由于运动维度低,基于现有的执行机构,二维无侧移运动自然是微分平坦的;三维无侧移质点运动在线速度不变的情况下是微分平坦的,而运动刚体的主轴与速度矢量的相对方位是进行三维无侧移运动动力学建模与控制的关键。
6.控制律设计:对三维无侧移运动的微分平坦输出进行设计,得到了一种恒速质点运动生成轨迹的新方法;对二维无侧移运动模型的微分平坦输出进行设计,可以实现统一的位姿镇定和轨迹跟踪控制律;进行路径跟踪时,设计恰当的微分平坦输出,使用精确线性化方法推导了路径跟踪控制律;在仅有视线角度测量信息的条件下,改造现有的平面比例导引律,增加对线速度的控制,实现了具有能量优化特性的位姿控制。
全文的研究采用了代数方法与几何方法结合的思想、空间分解的思想,深化了任务空间上多种运动形式的控制和规划问题的研究,为多自由度机械臂全任务空间控制问题和飞行器制导控制一体化问题的最终解决创造了条件。
Task-space is a typical configuration space of mechanical systems, whose geometrical correspondence is the Lie-group SE(3). The control problem on task space covers a variety of cases under research. However, due to the complexity of the space structure, theoretical research on task-space control advances slowly while being away from engineering applications. It is the defects of current research and the demands from engineering applications that initiates the research in this thesis, which focuses on two key issues of task-space control: the computation burden brought by matrix description of Lie-groups, and the no-literal-drift motion control.
The first half of this thesis aims at releasing the computation burden within control schemes, in seeking solution for whole task space control. Research on attitude control and posture control were carried out via the usage of quaternion algebra as representing tools, aiming at control schemes with both computational efficiency and utilization of task- space's geometric structures. The main achievements lay in the following three aspects.
1. The Lie-group structures of unit quaternions and normalized dual quaternions were formulated. Thereafter, generalized proportional-derivative(PD) control laws were developed for attitude control and posture control respectively, aiding by the Lie-algebras of the unit-quaternion Lie-group (Q_u) and the dual-quaternion Lie-group (DQ_u). With proper handling of the double-equilibrium problem brought by quaternion-like representations, both regulation and trajectory tracking control were realized for fully-actuated models.
2. Research on optimal attitude control and rotational trajectory design were carried out using unit quaternion. In the case study carried on the reorientation problem of free-floating space robot(FFSR), orientation control that can optimizing energy was achieved utilizing the Lie-algebra of Q_u and DQ_u. In discussing the two-point rotating interpolation problem with minimum acceleration, approximate analytical solutions were proposed using polynomial splines.
3. Research on control of multi-body mechanisms was carried out based on the notion of using dual-quaternion in posture control. The case study of multi-freedom manipulator shows how to combine a specific system model with the generalized PD law within the structure of the dual-quaternion Lie-Group DQ_u. Regulation law in whole task space was deduced, inducing the notation double task space control which leads to a control law for FFSR with minimum disturbance for the basement.
On the other hand, observing driftless motion which has aircrafts as the typical example, the model's characteristics makes it difficult to perform posture control, leading to the research on two sub-problems: trajectory planning of particles, and model analysis of driftless motion. Also, as a simplified version of driftless motion, control of ground mobile robots calls for a control scheme depending merely on angle detection. The latter part of the thesis has been focused on related problems of driftless motion control, arriving at achievements in three aspects.
4. Achievements in real-time trajectory planning: Focusing on the typical motion of moving particles with constant speed, combining with the notion of aircraft guidance, using the information brought by the variation of line-of-sight(LOS)'s orientation, the literal accelaration was calculated and real-time trajectory planning was accomplished. Description of the LOS's orientation can be performed by Lie-group or twist, yielding two algorithms for real-time trajectory planning. The two algorithms can also be used as guidance laws, which can fix the defects in traditional method where decoupling of channels were demanded, and partly meet the demands in multi-constraints guidance.
5. Achievements in model analysis: The analysis of driftless motion points out that, the degree-of-freedom(DOF) of motion affects motion particularity. With the known actuators, 2-dimensional(2D) driftless motion is naturally differential flat due to its minor DOFs. For 3-dimensional(3D) driftless motion, the kinematic model is differential flat only when the longitudinal velocity is constant, and the key for dynamic modeling and control is the relative rotation between the body axis and the velocity vector.
6. Achievements in control law design: A design on the flat output of 2D driftless motion results in a unified control law for both posture stabilization and trajectory tacking. When performing path following, the system was simplified to be 1-dimensional; With proper design of flat output, control law can be developed via exact linearizing. When only angle measurement of LOS was available, revising existing 2D guidance law, adding a control on the longitudinal velocity, posture stabilization with certain optimization was developed. For 3D driftless motion, design on the flat output, a new algorithm for trajectory generation is achieved for particle motion with constant speed.
The idea of combining algebraic method with geometric method, together with the skill of space decomposition, was utilized in the thesis, which deepens the research on control and trajectory planning of various kind of motion in task-space, and pave the way for a final solution to both the control of manipulators in whole task space and integrated guidance-control of aircrafts.
引文
[1]Murray R M.Control in an Information Rich World:Report of the Panel on Future Directions in Control,Dynamics,and Systems[M].the Society for Industrial and Applied Mathematics,2003
[2]Olfati-Saber R.Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[D].PH.D thesis,Massachusetts Institute of Technology,2001
[3]Goldstein H.Classical Mechanics[M].Addison-Wesley Publishing Company,1980
[4]Arnold V I.Mathemathical Methods of Classical Mechanics[M].New York:Springer-Verlag,1989,second edition,translated by K.Vogtmann and A.Weinstein.
[5]Boothby W M.An Introduction to Differentiable Manifolds and Riemannian Geometry [M].California:Academic Press,1986
[6]Marsden J E,Ratiu T S.Introduction to Mechanics and Symmetry[A].New York,NY:Springer Verlag,1994
[7]Bullo F,Lewis A D.Geometric Control of Mechanical Systems:Modeling,Analysis,and Design for Simple Mechanical Control Systems[M].NewYork:Springer,2004
[8]Liu G,Lou Y,Li Z X.Singularities of Parallel Manipulators:A Geometric Treatment [J].IEEE Transactions on Robotics and Automation.2003,19(4):579-594
[9]Boyd S,Barratt C.Linear Controller Design:Limits of Performance[M].Prcntice-Hall,1991
[10]Bloch A M,Crouch P E.Nonholonomic control systems on Riemannian manifolds[J].SIAM Journal on Control and Optimization.1995,33(1):126-148
[11]Lewis A D.Simple Mechanical Control Systems with Constraints[J].IEEE Transactions on Automatic Control.2000,45(8):1420-1436
[12]Lewis A D,Murray R M.Configuration controllability of simple mechanical control systems[J].SIAM Journal on Control and Optimization.1997,35(3):766-790
[13]Bullo F,Lynch K M.Kinematic Controllability for Decoupled Trajectory Planning in Underactuated Mechanical Systems[J].IEEE Transactions on Robotics and Automation.2001,17(4):402-412
[14]Bullo F.Trajectory Design for Mechanical Control Systems-From Geometry to Algorithms [J].European Journal of Control.2004
[15]Levi M.Geometry of Kapitsa's potentials[J].Nonlinearity.1998,11(5):1365-1372
[16]Liu W.An approximation algorithm for nonholonomic systems[J].SIAM Journal on Control and Optimization.1997,35(4):1328-1365
[17] Bullo F, Lewis A D. Reduction, linearization, and stability of relative equilibria for mechanical systems on Riemannian manifolds[J]. 2005
[18] Bullo F. Stabilization of Relative Equilibria for Underactuated Systems on Riemannian Manifolds[J]. Preprint submitted to Elsevier Preprint. 2005
[19] Bullo F, Cerven W T. On Trajectory Optimization for Polynomial Systems via Series Expansions [C]. Proceedings of the 39th IEEE Conference on Decision and Control. Sydney, Australia, 2000, 772-777
[20] Cortes J, Martinez S, Bullo F. On Nonlinear Controllability and Series Expansions for Lagrangian Systems With Dissipative Forces[J]. IEEE Transaction on Automatic Control. 2002, 47(8): 1396-1401
[21] Leonard N E, Krishnaprasad P S. Motion Control of Drift-Free Left-Invariant Systems on Lie Groups[J]. IEEE Transactions on Automatic Control. 1995, 40(9)
[22] Bullo F, Leonard N E, Lewis A D. Controllability and motion algorithms for underactuated Lagrangian systems on Lie groups[J]. IEEE Transactions on Automatic Control. 2001, 45(8):1437-1454
[23] Biggs J, Holderbaum W, Jurdjevic V. Singularities of Optimal Control Problems on Some 6-D Lie Groups[J]. IEEE Transactions on Automatic Control. 2007, 52(6):1027-1038
[24] Walsh G C, Montgomery R, Sastry S S. Optimal Path Planning on Matrix Lie Groups[C]. Proceedings of the 33rd Conference on Decision and Control. Lake Buena Vista, FL, 1994, 1258-1263
[25] Justh E W, Krishnaprasad P S. Natural frames and interacting particles in three dimensions[C]. Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005. Seville, Spain, 2005, 2481-2486
[26] Bloch A, Crouch P, Ratiu T. Sub-Riemannian optimal control problems[J]. Fields Institute Communications. 1994, 3:35-48
[27] Hussein I I, Bloch A M. Dynamic Interpolation on Riemannian Manifolds: An Application to Interferometric Imaging[C]. Proceedings of the 2004 American Control Conference. Boston, Massachusetts, USA, 2004, 685-690
[28] Hussein 11, Bloch A M. Dynamic Interpolation on Riemannian Manifolds: An Application to Interferometric Imaging Formations: The Nonlinear Case[C]. Proceedings of the 2005 American Control Conference. Portland,OR,USA, 2005, 2391-2396
[29] Hussein I I, Bloch A M. Optimal Control of Under-actuated Systems with Application to Lie Groups[C]. Proceedings of the 2005 American Control Conference. Portland,OR,USA, 2005, 1472-1477
[30] Nordkvist N, Bullo F. Control algorithms along relative equilibria of underactuated Lagrangian systems on Lie groups[C]. New Orleans, LA. 2007
[31]Taylor R H.Planning and Execution of Straight Line Manipulator Trajectories[J].IBM Journal of Research and Development.1982,23(4):424-436
[32]Lloyd J,Hayward V.Real-time Trajectory Generation Using Blend Functions[C].IEEE International Conference on Robotics and Automation.Sacramento,Calif.,1991,784-789
[33]Zha X F.Optimal Pose Trajectory Planning for Robot Manipulators[J].Mechanism and Machine Theory.2002,37(10):1063-1086
[34]Nejat G,Benhabib B.High-precision task-space sensing and guidance for autonomous robot localization[C].Proceedings of the 2003 IEEE International Conference on Robotics and Automation(ICRA).Talpei,Taiwan,2003,1527-1532
[35]Nejat G,Benhabib B,Membre A.Active Task-Space Sensing and Localization of Autonomous Vehicles[C].Proceedings of the 2005 IEEE International Conference on Robotics and Automation(ICRA).2005,3770-3775
[36]Nejat G,Benhabib B.Line-of-sight Task-space Sensing Methodology for the Localization of Robotic End-effectors[J].IEEE/ASME Transactions on Mechatronics.2006,11(2):225-232
[37]Zefran M.Continuous Methods for Motion Planning[D].PH.D thesis,University of Pennsylvania,1996
[38]Park F C,Ravani B.Smooth Invariant Interpolation of Rotations[J].ACM Transactions on Graphics.1997,16(3):277-295
[39]Kang I G,Park F C.Cubic Spline Algorithms for Orientation Interpolation[J].International Journal of Numerical Methods in Engineering.1999,46:45-64
[40]Zefran M,Kumar V,Croke C.Choice of Riemannian metrics for rigid body kinematics [C].Proceedings of The 1996 ASME Design Engineering Technical Conference and Computers in Engineering Conference.Irvine,California,1996
[41]Belta C,Kumar V.Euclidean Metrics for Motion Generation on SE(3)[J].Journal of Mechanical Engineering Science Part C.2002,216(C1):47-60
[42]Zefran M,Kumar V.Interpolation schemes for rigid body motions[J].Computer aided design.1998,30:179-189
[43]Belta C,Kumar V.On the Computation of Rigid Body Motion[J].Electronic Journal of Computational Kinematics.2002,1(1)
[44]Belta C,Kumar V.An SVD-Based Projection Method for Interpolation on SE(3)[J].IEEE Transactions on Robotics and Automation.2002,18(3):334-345
[45]Sharpe R W.Differential Geometry:Cartan's generalization of Klein's Erlangen Program[M].New York:Springer-Verlag,1996
[46]H(u|¨)per K,Leite F S.On the Geometry of Rolling and Interpolation Curves on S_n,SO_n and Graβmann Manifolds[R].Tech.Rcp.Technical Report SISSA 56/2005/M,2005 2005
[47]Shen Y s,Hiiper K,Leite F S.Smooth Interpolation of Orientation by Rolling and Wrapping for Robot Motion Planning[C].Proceedings of the 2006 IEEE International Conference on Robotics and Automation(ICRA).Orlando,Florida,2006,113-118
[48]Ge Q J,B.Ravani.Computer Aided Geometric Design of Motion Interpolants[J].ASME Journal of Mechanical Design.1994,116(3):756-762
[49]Ge Q J,B.Ravani.Geometric construction of Bezier motions[J].ASME Journal of Mechanical Design.1994,116(3):749-755
[50]Purwar A,Ge Q J.On the effects of weights in rational motion design[C].Proceedings of DETC'03 ASME 2003 Design Engineering Technical Conferences and Computers and Information in Engineering Conference.Chicago,Illinois,USA,2003
[51]Purwar A,Ge Q J.On the Effect of Dual Weights in Computer Aided Design of Rational Motions[J].Journal of Mechanical Design.2005,127:967-972
[52]Kuo C-Y,Wang S-P T.Robust Position Control of Robotic Manipulator in Cartesian Coordinates[J].IEEE Transactions on Robotics and Automation.1991,7(5):653-659
[53]Caccavale F,Natale C,Villani L.Task-space Tracking Control Without Velocity Measurements [C].Proceedings of the 1999 IEEE International Conference on Robotics and Automation.Detroit,MI,USA,1999,vol.1,512-517
[54]Liu C,Cheah C C.Task-Space Adaptive Setpoint Control for Robots With Uncertain Kinematics and Actuator Model[J].IEEE Transactions on Automatic Control.2005,50(11):1854-1860
[55]Takegaki M,Arimoto S.A new feedback method for dynamic control of manipulators [J].ASME Journal of Dynamic Systems,Measurement,and Control.1981,102:119-125
[56]Kelly R.Regulation of manipulators in generic task space:An energy shaping plus damping injection approach[J].IEEE Transactions on Robotics and Automation.1999,15(2):381-386
[57]Cheah C C,Liaw H C.Inverse Jacobian Regulator With Gravity Compensation:Stability and Experiment[J].IEEE Transactions on Robotics.2005,21(4):741-747
[58]H(u|¨)per K,Shen Y.Optimal trajectory planning of manipulators subject to motion constraints[C].Proceedings of 12th International Conference on Advanced Robotics.2005,9-16
[59]Mochiyama H.Task space control for the whole arm[C].Proceedings of Conference on 2000 IEEE/RSJ International Intelligent Robots and Systems(IROS 2000).Takamatsu,Japan,2000,vol.3,1772-1777
[60]Seraji H.Configuration Control of Redundant Manipulators:Theory and Implementation [J].IEEE Transaction on Robotics and Automation.1989,5(4):472-490
[61]Seraji H,Long M,Lee T.Conflguration Control of 7 DOF Arms[C].Proceedings of the 1991 IEEE International Conference on Robotics ans Automation.Sacramento,California,1991,1195-1200
[62]Seraji H.Configuration Control of Rover-mounted Manipulators[C].Proceedings of 1995 IEEE International Conference on Robotics and Automtion.1995,2261-2266
[63]Asama H,Sato M,Bogoni L,Kactsu H.Development of an Omni-Directional Mobile Robot with 3 DOF Decoupling Drive Mechanism[C].Proceedings of the 1995International Conference on Robotics and Automation.Nagoya,Aichi,Japan,1995,1925-1390
[64]Murray R M,Sastry S S.Nonholonomic Motion Planning:Steering Using Sinusoids [J].IEEE Transactions on Automatic Control.1993,38(5):700-716
[65]Pomet J-B.Explicit Design of Time-varying Stabilizing Control Laws for a Class of Controllable Systems without Drift[J].Systems & Control Letters.1992,18:147-158
[66]Astolfi A.Discontinuous Control of Nonholonomic Systems[J].Systems & Control Letters.1996,27(1):37-45
[67]Samson C.Control of Chained Systems Application to Path Following and Time-Varying Point-Stabilization of Mobile Robots[J].IEEE Transactions on Automatic Control.1995,40(1):64-77
[68]Chwa D.Sliding-Mode Tracking Control of Nonholonomic Wheeled Mobile Robots in Polar Coordinates[J].IEEE Transactions on Control Systems Technology.2004,12(4):637-644
[69]Lee T-C,Song K-T,Lee C-H,Teng C-C.Tacking Control of Unicycle-Modeled Mobile Robots Using a Saturation Feedback Controller[J].IEEE Transactions on Control Systems Teclmology.2001,9(2):305-318
[70]Dixon W E,Jiang Z P,Dawson D M.Global Exponential Setpoint Control of Wheeled Mobile Robots:a Lyapunov Approach[J].Automatica.2000,36:1741-1746
[71]Oriolo G,Luca A D,Vendittelli M.WMR Control Via Dynamic Feedback Linearization:Design,Implementation,and Experimental Validation[J].IEEE Transactions on Control Systems Technology.2002,10(6):835-852
[72]Sun S,Cui P.Path Tracking and a Practical Point Stabilization of Mobile Robot[J].Robotics and Computer-Integrated Manufacturing.2004,20:29-34
[73]Khaneja N,Brockett R.Dynamic Feedback Stablization of Nonholonomic Systems [C].Proceedings of the 38th Conference on Decision & Control.Phoenix,Arizona USA,1999,1640-1644
[74]Fliess M.Flatness and Defect of Nonlinear Systems:Introductory Theory and Examples [J].International Journal of Control.1995,61(6):1327-1361
[75]Rouchon P,Fliess M.Flatness,motion planning and trailer systems[C].Proceedings of the 32nd IEEE Conf.Decision and Control.San Antonio,TX,1993,2700-2705
[76]P.Rouchon,M.Fliess,J.Levine,P.Martin.Flatness and Motion Planning:the Car with n Trailers[J].1992
[77]Yuan P-J,Chern J-S.Ideal Proportional Navigation[J].Journal of Guidance,Control,and Dynamics.1992,15(5):1161-1165
[78]Yang C-D,Yang C-C.Analytical solution of generalized 3D proportional navigation [C].Proceedings of the 34th IEEE Conference on Decision and Control,1995.New Orleans,LA,USA,1995,3974-3979
[79]Cloutier J R,Evers J H,Feeley J J.Assessment of air-to-air missile guidance and control technology[J].IEEE Control Systems Magazine.1989,9(6):27-34
[80]Ha I-j,Chong S.Design of a CLOS Guidance Law via Feedback Linearization[J].IEEE Transactions on Aerospace and Electronic Systems.1992,28(1):51-63
[81]Lee Y-I,Ryoo C-K,Kim E.Optimal Guidance with Constraints on Impact Angle and Terminal Acceleration[C].AIAA Guidance,Navigation,and Control Conference and Exhibit.Austin Texas,2003
[82]Ryoo C-K,Cho H,Tahk M-J.Cloud-form Solutions of Optimal Guidance with Terminal Impact Angle Constraint[C].Proceedings of 2003 IEEE Conference on Control Applications.2003,504-509
[83]Balakristman S N,Stansbery D T.Analytical Guidance Laws and Integrated Guidance /Autopilot for Homing Missiles[C].Second IEEE Conference on Control Applications.Vancouver B C,1993,13-16
[84]Menon P K,lmeyer E J O.Integrated Design of A gile Missile Guidance and Autopilot Systems[J].Control Engineering Practice.2001,(9):1095-1106
[85]Bullo F,Murray R.Proportional derivative(PD) control on the Euclidean group[C].1995 European Control Conference.Rome,Italy,1995
[86]Funda J,Taylor R H,paul R P.On Homogeneous Transforms,Quaternions,and Computational Efficiency[J].IEEE Transactions on Robotics and Automation.1990,6(3):382-388
[87]Funda J,Paul R P.A Computational Analysis of Screw Transformations in Robotics[J].IEEE Transactions on Robotics and Automation.1990,6(3):348-356
[88]Wen J T-Y,Kreutz-Delgado K.The Attitude Control Problem[J].IEEE Transactions on Automatic Control.1991,36(10):1148-1162
[89]Joshi S M,Kelkar A G,Wen J T-Y.Robust Attitude Stabilization of Spacecraft Using Nonlinear Quaternion Feedback[J].IEEE Transactions on Automatic Control.1995,40(10):1800-1803
[90]Fragopoulos D,Innocenti M.Stability Considerations in Quaternion Attitude Control using Discontinuous Lyapunov Functions[J].IEE Proc-Control Theory and Application.2004,151(3):253-258
[91]Bottema O,Roth B.Theoretical Kinematics[M].New York:North-Holland Publishing Company,1979
[92]Wu Y X,Hu X P,Hu D W,Lian J X.Strapdown Inertial Navigation System Algorithms Based on Dual Quaternions[J].IEEE Transactions on Aerospace and Electronic Systems.2005,41(1):110-132
[93]Kim M-J,Kim M-S.A Compact Differential Formula for the First Derivative of a Unit Quaternion Curve[J].Journal of Visualization and Computer Animation.1996,7(1):43-57
[94]Koditschek D E.The application of total energy as a Lyapunov function for mechanical control systems[A].Krishnaprasad P S,Marsden J E,Simo J C,(Editors)Dynamics and Control of Multibody Systems,AMS,1989,vol.97.131-157
[95]Vafa Z,Dubowsky S.On the dynamics of space manipulator using the virtual manipulator with application to path planning[J].Journal of the Astronautical Sciences.1990,38(4):441-472
[96]Walsh G C,Sastry S S.On reorienting linked rigid bodies using internal motions[J].IEEE Transactions on Robotics and Automation.1995,2(1):139-146
[97]Fernandes C,Gurvits L,Li Z.Near optimal nonholonomic motion planning for a system of coupled rigid bodies[J].IEEE Transactions on Automatic Control.1994,39(3):450-463
[98]Qu Z H,Wang J,Plaisted C E.A New Analytical Solution to Mobile Robot Trajectory Generation in the Presence of Moving Obstacles[J].IEEE Transactions on obotics.2004,20(6):978-993
[99]Goddard J J S.Pose and Motion Estimation from Vision Using Dual Quaternionbased Extended Kalman Filtering[D].PH.D thesis,The University of Tennessee,1997
[100]Daniilidis K.Hand-Eye Calibration Using Dual Quaternions[J].The International Journal of Robotics Research.1999,18(3):286-298
[101]Dooley J R,McCarthy J M.On the Geometric Analysis of Optimum Trajectories for Cooperating Robots Using Dual Quaternion Coordinates[C].Proceedings of the 1993 International Conference on Robotics and Automation.1993:1031-1036
[102]Dooley J R,McCarthy J M.Spatial Rigid Body Dynamics using Dual Quaternion Components[C].Proceedings of the 1991 International Conference on Robotics and Automation.Sacramento California,1991,90-95
[103]Hsia L M,Yang A T.On the Principle of Transference in Three-Dimensional Kinematics [J].ASME Journal of Mechnical Design.1981,103:652-656
[104]Kalmar-Nagy T a,D'Andrea R,Ganguly P.Near-optimal Dynamic Trajectory Generation and Control[J].Robotics and Autonomous Systems.2004,46(1):47-64
[105]Abrams S,K.Allen P.Swept volumes and their use in viewpoint computation in robot work-cells[C].1995 IEEE International Symposium on Assembly and Task Planning.Pittsburgh,Pennsylvania,1995,188-193
[106]Yuan J S.Closed-loop manipulator control using quaternion feedback[J].IEEE Journal of Robotics and Automation.1988,4(4):434-440
[107]Xian B,Queiroz M S d,Dawson D,I.Walker.Task-space Tracking Control of Robot Manipulators via Quaternion Feedback[J].IEEE Transactions On Robotics and Automation.2004,20(1):160-167
[108]Tyan F.An Unified Approach to Missile Guidance Laws:A 3D Extension[C].Proceedings of the American Control Conference.Anchorage,AK,2002,1711-1716
[109]Murray R M,Li Z X,Sastry S S.An Mathematical Introduction to Robotic Manipulation [M].CRC Press,1994
[110]Rossberg K.A First Course in Analytical Mechanics[M1.John Wiley & Sons,Inc.,1983
[111]Lee J G,Han H S,Kim Y J.Guidance Performance Analysis of Bank-To-Turn(BTT)Missiles[C].Proceedings of the 1999 IEEE International Conference on Control Applications.Kohala Coast-Island of Hawai'i,Hawai'i,USA,1999,991-996
[112]Nejat G,Beahabib B.A Guidance-based Motion-planning Methodology for the Docking of Autonomous Vehicles[J].Journal of Robotic Systems.2005,22(12):779-793
[113]Fliess M.A Lie-B(a|¨)cklund Approach to Equivalence and Flatness of Nonlinear Systems [J].IEEE Transactions on Automatic Control.1999,44(5):922-967
[114]Nieuwstadt M V,Rathinam M,Murray R M.Differential flatness and absolute equivalence[J].SIAM Journal of Control and Optimization.1998,36(4):1225-1239
[115]Nieuwstadt M V,Murray R M.Real Time Trajectory Generation For Differentially Flat Systems[J].International Journal of Robust Nonlinear Control.1997,8:995-1020
[116]Rathinam M,Murray R M.Differential flatness of two one-forms in arbitrary number of variables[J].Systems & Control Letters.1999,36:317-326
[117]Egerstedt E,Hu X,Stotsky A.Control of Mobile Platform Using a Virtual Vehicle Approach[J].IEEE Transactions on Automatic Control.2001,46(11):1777-1782
[118]Kang W,Xi N,Tan J.Analysis and Design of Non-time Based Motion Controller for Mobile Robots[C].ICRA.Detroit,Michigan,1999,2964-2969
[119]Lizarralde F,Wen J T.Attitude Control Without Angular Velocity Measurements:A Passivity Approath[J].IEEE Transactions on Automatic Control.1996,41(3):468-472
[120]Tayebi A.Unit quaternion observer based attitude stabilization of a rigid spacecraft without velocity measurement[C].45th IEEE Conference on Decision and Control.San Diego,CA,USA,2006
[121]Chou J C K.Quaternion Kinematic and Dynamic Differential Equations[J].IEEE Transactions on Robotics and Automation.1992,8(1):53-64
[122]Andreis D,Canuto E S.Orbit dynamics and kinematics with full quaternions[C].Proceeding of the 2004 American Control Conference.Boston,Massachusetts,2004,3660-3665
[123]Lewis M A,Tan K-H.High Precision Formation Control of Mobile Robots Using Virtual Structures[J].Autonomous Robots.1997,4(4):387-403
[124]Balch T,Arkin R C.Behavior-based formation control for multirobot teams[J].Robotics and Automation,IEEE Transactions on.1998,14(6):926-939
[125]Desai J P,Ostrowski J P,Kumar V.Modeling and Control of Formations of Non-holonomic Mobile Robots[J].IEEE Transactions on Robotics and Automation.2001,17(6):905-90S
[126]Belta C,Kumar V.Optimal Motion Generation for Groups of Robots:A Geometric Approach[J].Journal of Mechanical Design.2004,126(63):63-70
[127]Brockett R W.Asymptotic stability and feedback stabilization[A].In:Brockett RW,Millman RS,Sussmann HJ(eds).Differential Geometric Control Theory.Boston,MA:Birkh(a|¨)user,1983:181-191
[128]程代展.非线性系统的几何理论[M].北京:科学出版社,1988
[129]Baillieul J.Stable average motions of mechanical systems subject to periodic forcing [A].In:Enos MJ(ed).Dynamics and Control of Mechanical Systems:The Falling Cat and Related Problems,Field Institute Communications,1993:1-23
[130]夏小华,高为炳.非线性系统控制及解耦[M]:科学出版社,1993
[131]陈克俊,赵汉元.一种适用于攻击地面固定目标的最优再入机动制导律[J].宇航学报,1994,15(1):1-7
[132]赵汉元.飞行器再入动力学和制导[M].长沙:国防科技大学出版社,1997
[133]余杨政,冯承天.物理学中的几何方法[M].北京:高等教育出版社,1998
[134]赵东标,查选芳,王珉,朱剑英.基于螺旋理论的机械手最优位姿轨迹规划方法研究[J].航空学报,1999,20(4):331-334
[135]董文杰,霍伟.链式系统的轨迹跟踪控制[J].自动化学报,2000,26(3):310-316
[136]王越超,景兴建.轮式移动机器人控制[J].机器人,2000,22(7):724-729
[137]晁红敏,胡跃明.动态滑模控制及其在移动机器人输出跟踪中的应用[J].控制与决策,2001,16(5):565-568
[138]陈微桓.微分流形初步[M].北京:高等教育出版社,2001
[139]连葆华,崔平远,崔祜涛.高速再入飞行器的制导与控制系统设计[J].航空学报,2002,23(2):115-119
[140]周荻.寻的导弹新型导引规律[M].北京:国防工业出版社,2002
[141]王栋耀,马旭东,戴先中.非时间参考的移动机器人路径跟踪控制[J].机器人,2004,26(3):198-203
[142]查旭,崔平远,常伯浚.攻击固定目标的飞行器制导控制一体化设计[J].宇航学报,2005,26(1):13-18
[143]洪炳镕,柳长安,刘宏.自由飞行空间机器人运动控制及仿真[M].北京:国防工业出版社,2005
[144]于进勇,张友安,顾文锦.反舰导弹航向平面控制导引一体化设计[J].飞行力学,2005,23(3):67-69
[145]虞铭财,杨勋年,汪国昭.高阶连续的单位四元数插值曲线[J].计算机辅助设计与图形学学报,2005,17(3):437-441
[146]Han D P,Wei Q,Li Z X.Path Following of Mobile Robots Using a Virtual Vehicle Approach [C].Proceedings of the 25th Chinese Control Conference.Harbin,Hcilongjiang,2006:1533-1537
[147]徐文福,詹文法,梁斌,李成,强文义.自由漂浮空间机器人系统基座姿态调整路径规划方法的研究[J].机器人,2006,28(3):291-296
[148]孙未蒙,郑志强.多约束条件下对地攻击的最优制导律[J].兵工学报,2008,29(5):567-571
[149]Yoshida K.The Spacedyn:a MATLAB Toolbox for Space and Mobile Robots[EB/OL].http://www.astro.mech.tohoku.ac.jp/spacedyn/