[精品论文]Acceleration Analysis of Rigid Body and Its Application for.doc

精品论文Acceleration Analysis of Rigid Body and Its Application forShip-Based Stabilized Platform SystemZHAO Tieshi1,2, LIU Xiao1,2, YUAN Feihu1,2, GENG Mingchao1,25(1. Parallel Robot and Mechatronic System Laboratory of Hebei Province,Yanshan University, HeBei QinHuangDao 066004;2. Key Laboratory of Advanced Forging & Stamping Technology and Science of Ministry ofNational Education, Yanshan University, HeBei QinHuangDao 066004)Abstract: The traditional representation of acceleration of a rigid body is given in terms of the angular10acceleration and linear acceleration of a point attached to the rigid body. Since this representation has no coordinate invariance, the acceleration transformation and mapping of a multi-rigid-body system are complicated. In this paper, the physical meaning of the time derivative of a twist is investigated. Itreveals that the rigid-body acceleration comprises the angular acceleration and tangent acceleration of apoint which is attached to the rigid body and instantaneously coincident with the origin of frame in use.15Their composition presents a six-dimensional representation of the rigid-body acceleration, which is verified to be of coordinate invariance. Based on the representation, the transformation of the rigid-body accelerations is performed conveniently, and the corresponding formula of composition accelerations of one rigid body relative to any other bodies in a multi-rigid-body system is presented. The acceleration mapping that maps the coordinates of a point to its acceleration is also developed. The20method is then extended to the application of a ship-based stabilized platform system and an algorithm of velocity and acceleration correction for the helicopter landing safely is presented. The method isverified to be effective by analyzing the virtual prototype of the ship-based stabilized platform system. This paper builds a bridge for the six-dimensional rigid-body acceleration from theory achievements to practical application.25Keywords: Parallel Robots; Kinematics; Rigid-Body Acceleration; Ship-Based Stabilized Platform0IntroductionWith the development of marine science, the ship-based stabilized platform became a hot area of research in resent years 1-3. The role of a ship-based stabilized platform is to provide a30stable working platform for the helicopter 4-5, as shown in Fig.1. As the coupling action amongthe helicopter, ship and stabilized platform, an algorithm of the velocity and acceleration correction is needed to make the helicopter landing safely. Since the ship, stabilized platform, and helicopter constitute a multi-rigid-body system, it is complicated to calculate, transform, and compose the relative accelerations of the rigid bodies by means of the vector algebra.35The kinematics of the rigid bodies is most commonly formulated in terms of the vector algebra, in which rotation and translation are described individually. The Acceleration of a rigid body is described in terms of the angular acceleration and linear acceleration of any point attached to the rigid body. Since this representation have no coordinate invariance, a complicated work arises when applying the traditional description to the acceleration analysis of a multi-rigid-body40system 6. Therefore, Lagrange method 7-9 has been applied widely to the dynamics of themulti-rigid-body systems since there are no accelerations needed. However, Lagrange method transfers the complicated work to the derivation of partial differential of a Lagrangian 10.Modern mathematic tools, such as the spatial operator algebra 11, Grassmann-Cayley algebra12, geometric algebra 13, screw algebra 14, Lie groups and Lie algebras 15-16 have been45introduced to the rigid body kinematics in order to obtain more concise formulations. Based on the homogeneous transformation, Lie algebra and screw theory, Aspragathos 17 presented threeFoundations: National Science Foundation of China Grant(No.50975244)Brief author introduction:ZHAO Tieshi, (1963-) is currently a professor with the Department of Mechatronics Engineering, Yanshan University. His research interests include parallel mechanisms, stabilized platforms, sensor technology, and robotics technology. E-mail: tszhao- 22 -methods to formulate the kinematic equation of the robot arms, and proved that the screw theory and Lie algebra-based methods were more cost effective as the number of the robot degrees of freedom increases. The pioneering treatise by Ball 18 laid the foundations of the screw theory; the50outstanding works by Hunt 19, Duffy 20 and Phillips 21 promoted modern applications of thescrew theory; and a great deal of the work has been contributed to the robot kinematics 22-24. Nevertheless, until recent years, the most of work on the screw theory has been confined to the velocity analysis 25, static analysis 26, geometric analysis 27, and synthesis 28. Only several contributions have made some progress in extending screw theory into the acceleration analysis of55rigid bodies. Sugimoto 29 obtained the acceleration motor of the end effector of a manipulator bycalculating the time derivative of the velocity expression of the end effector. Bokelberg 30 discussed the differential screw by analyzing an infinitesimal displacement of a velocity screw and Ridley 31 further indicated the acceleration center of a rigid body in a general spatial motion. Martinez and Duffy 32-33 investigated the composition of the reduced acceleration states and60reduced jerk states by individually calculating the time derivatives of three-dimensional vectors denoted the angular velocities and linear velocities, and indicated that reduced acceleration states of the rigid bodies are coordinate invariant. However, the physical meaning of the time derivative of a twist is still required; it has been known that the time derivative of a twist is different from the traditional representation of the rigid-body accelerations.65In this paper, the physical meaning of the time derivative of a twist is investigated. It reveals that the rigid-body acceleration comprises of the angular acceleration and tangent acceleration of a point which is attached to the rigid body and instantaneously coincident with the origin of frame in use. The linear component of the rigid-body acceleration is termed the linear acceleration of a rigid body. The angular acceleration and linear acceleration comprises a six-dimensional70representation of the rigid-body acceleration, which is verified to be of coordinate invariance.Then the transformation of the rigid-body acceleration is performed, which results in a composition accelerations formula of one rigid body relative to any other bodies in a multi-rigid-body system. The acceleration mapping that maps the coordinates of a point to its rigid-body acceleration is also developed. The method can be extended to the application of a75ship-based stabilized platform system and an algorithm of velocity and acceleration correction for the helicopter landing safely is presented. Finally, the virtual prototype of the ship-based stabilized platform system is analyzed.Fig. 1 The Ship-based stabilized platform and helicopter801Physical meaning of time derivative of a twistIn this section, the physical meaning of the time derivative of a twist is investigated, which reveals that the rigid-body acceleration comprises the angular acceleration and tangent acceleration of a point which is attached to the rigid body and instantaneously coincident with the85origin of frame in use.Let a and b denote the reference frames attached to rigid bodies A and B, respectively.The acceleration of rigid body B can be obtained by differentiating the twist, aV= g&g1 , of rigidbody B in terms of frame a,a 11ab ab ab Ra p Aab = &g&ab gab + g& ab g& ab ,(1)90where, g=ababis the homogeneous representation of the configuration of rigidab 01 a a a body B relative to frame a. Setting a A = ab ab , the following equation is obtained byab 00 expanding (1),a T Tab = R&&ab Rab + R& ab R& ab ,where, Rab is the rotation matrix of rigid body B with respect to frame a. It can be directly95identified with the angular acceleration of rigid body B, asa = R& RTindicates the angularabab abvelocity of rigid body B in terms of frame a. Similarly, the linear component of the time derivative of the twist, which is termed the linear acceleration of a rigid body in this paper, can also be obtained from (1),abaaa a a aaab =&p&ab abpab abp& ab ,(2)100where, apab is the position vector of origin b. Sinceelement of se(3).a so(3) anda 3 ,a aabAabis anTraditionally, the acceleration of rigid body B with respect to frame a is described with the angular acceleration of rigid body B and the linear acceleration of a point attached to rigid body B.Let abe the vector representation of angular acceleration a ; a &p&the linear acceleration ofababba105point a which is attached to rigid body B and coincident with the origin of frame a.abba( a ，a &p& )gives a traditional representation of the acceleration of rigid body B with respect to110115frame a, which is a composition of two independent parameters. It has been observed that the time derivative of the twist has the same angular component and different linear component from the traditional representation of the acceleration of a rigid body. The following investigation into the physical meaning reveals that the time derivative of a twist is the unitary and completeabsix-dimensional representation of the rigid-body acceleration. Its linear component, a a , can be termed the linear acceleration of a rigid body, which is no longer the acceleration of a fixed point attached at the rigid body.The velocity of point a can be expressed from the transformation formulae of linearvelocities,a a a ap& aa =p& ab +abpba .The acceleration of point a can be given by differentiating the above equation,aaa a a ( a a )&p&aa =&p&ab +abpba +ababpba .Equation (2) can be simplified by using the above two equations and consideringa a120pba = pabas,a a a aaab =&p&aa abp& aa .(3)sa Let s denote the point attached to the twisting axis of rigid body B; a pthe vector fromaspoint s to point a ; a &p&given in Fig. 2.the acceleration of the point s relative to rigid body A, an illustration is125Fig. 2 The velocity and acceleration of a rigid body.From the screw theory, the velocity of point a can be expressed asa a a ap& aa = habab +abpsa (4)130where, hab is the pitch of the velocity. Using (4) gives,a aa ( a a )abp& aa =ababpsa .(5)It indicates thatinto (3) givesa ap&ab aa represents the centripetal acceleration of the pointa' . Substituting (5)aaa ( a a )aab =&p&aa ababpsa (6)135The above equation reveals that the linear acceleration of a rigid body represents the acceleration of a point that is currently passing through the origin of frame a without its centripetal acceleration component. That means the linear acceleration of a rigid body is the tangent acceleration of the point which is attached to rigid body B and instantaneously coincident with theorigin of frame a.140The acceleration of point accelerations asa' can also be given from the transformation formulae of linearaaa a a ( a a )&p&aa =&p&as +abpsa +ababpsa ,(7)awhere,&p&asis the acceleration of point s;a apab sa is the acceleration caused as the acceleratingrotation of the rigid body about the twisting axis;a ( a a p )is the centripetal accelerations145of point a . Substituting (7) into (6) yieldsaaa ab ab sa aaab =&p&as +abpsa .150155The above equation further indicates that the linear acceleration of a rigid body composes of the acceleration of the point attached to the twisting axis of the rigid body and the acceleration ofpoint a caused as the accelerating rotation of the rigid body about the twisting axis. These two components integrate into the tangent acceleration of point a , which is termed the linearacceleration of a rigid body in this paper. The composition of the angular acceleration and linearacceleration of a rigid body is an element of se(3), which is unitary and complete representation of the rigid-body acceleration. It can be used in six-dimensional transformations of rigid-body accelerations, and can be also directly applied to obtain the acceleration of any point attached to a rigid body. The corresponding formulae are given in the following sections.2Transformation of rigid-body accelerationsThe following four sections develop the transformation and mapping methods of the rigid-body accelerations, which build the bridges from concept to application of the rigid-body accelerations.160Equation (1) gives the acceleration of rigid body B relative to rigid body A in terms of frame a.Similarly, the acceleration of rigid body B in terms of frame b can be obtained by differentiatingthe twistbV= g1 g&, i.e.ab ababb 1 1Expanding the above equation yieldsAab = g& ab g& ab + gab &g&ab .(8)165b = R& T R&+ RT R&.(9)ab ab ab ab abIt can be proved to be a skew-symmetric matrix by differentiatingR RTab ab= I twice and definedas the angular acceleration of rigid body B described in terms of frame b. Expanding equation (8)and settingb a = RT a &p&+ R& T a p&(10)ab ab ab abab170Substitutingb &p&T a= R&p&ab ababandb p&T a= Rp&ab ababinto(1