云南22万高考生周日赶考 54%录取率高过去年

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The Connectivity of Boolean Satisfiability: Computational and Structural DichotomiesParikshit GopalanGeorgia Tech. parik@Phokion G.KolaitisIBM Almaden.kolaitis@Elitza N.ManevaUC Berkeley.elitza@ Christos H.PapadimitriouUC Berkeley.christos@July29,2006AbstractBoolean satisfiability problems are an important benchmark for questions about complexity,algorithms, heuristics and threshold phenomena.Recent work on heuristics,and the satisfiability threshold has centeredaround the structure and connectivity of the solution space.Motivated by this work,we study structural and connectivity-related properties of the space of solutions of Boolean satisfiability problems and establish various dichotomies in Schaefer’s framework.On the structural side,we obtain dichotomies for the kinds of subgraphs of the hypercube that can be induced by the solutions of Boolean formulas,as well as for the diameter of the connected components ofthe solution space.On the computational side,we establish dichotomy theorems for the complexity of the connectivity and-connectivity questions for the graph of solutions of Boolean formulas.Our results assert that the intractable side of the computational dichotomies is PSPACE-complete,while the tractable side-which includes but is not limited to all problems with polynomial time algorithms for satisfiability-is in P for the-connectivity question,and in coNP for the connectivity question.The diameter of components can be exponential for the PSPACE-complete cases,whereas in all other cases it is linear;thus,small diameter and tractability of the connectivity problems are remarkably aligned.The crux of our results is an expressibility theorem showing that in the tractable cases,the subgraphs induced by the solution space posses certain good structural properties,whereas in the intractable cases,the subgraphs can be arbitrary.1IntroductionIn1978,T.J.Schaefer[20]introduced a rich framework for expressing variants of Boolean satisfiability and proved a remarkable dichotomy theorem:the satisfiability problem is in P for certain classes of Boolean formu-las,while it is NP-complete for all other classes in the framework.In a single stroke,this result pinpoints the computational complexity of all well-known variants of S AT,such as-S AT,H ORN3-S AT,N OT-A LL-E QUAL -S AT,and-IN-S AT.Schaefer’s work paved the way for a series of investigations establishing dichotomiesfor several aspects of satisfiability,including optimization[6,8,14],counting[7],inverse satisfiability[13], minimal satisfiability[15],-valued satisfiability[5]and propositional abduction[9].Our aim in this paper is to carry out a comprehensive exploration of a different aspect of Boolean satisfiabil-ity,namely,the c onnectivity properties of the space of solutions of Boolean formulas.The solutions(satisfying assignments)of a given-variable Boolean formula induce a subgraph of the-dimensional hypercube, which we call the solution graph.We believe that connectivity properties of such graphs merit study in their own right,as they shed light on the structure of the solution space of Boolean formulas.Furthermore,in recent years the structure of the solution graph for random instances has been the main consideration at the basis of both algorithms for and mathematical analysis of the satisfiability problem[2,18,17,16].It has been conjectured for 3-S AT[17]and proved for8-S AT[19,3],that the solution space fractures as one approaches the critical region from below.This apparently leads to performance deterioration of the standard satisfiability algorithms,such as WalkSAT[21]and DPLL[1].It is also the main consideration behind the design of the survey propagation algorithm,which has far superior performance on random instances of satisfiability[17].This body of work has served as a motivation to us for pursuing the investigation reported here.While there has been an intensive study of the structure of the solution space of Boolean satisfiability problems for random instances,our work seems to be thefirst to explore this issue from a worst-case viewpoint.Our Results.Our work addresses the question:when does the solution graph of a Boolean formula have nice structure?To answer this question,one must clarify what is meant by nice structure.One can define it in terms of graph theoretic properties of the solution graph.We can ask what kinds of graphs are possible as solution graphs of a Boolean formula.One can focus on specific structural properties such as diameter of each component.Alternatively,once can view the Boolean formula as an implicit description of the solution graph,and study the computational complexity of algorithmic tasks such asfinding if the graph is connected. Surprisingly,we show that many of these properties,both structural and algorithmic,are remarkably aligned and result in the same dichotomies.We identify two broad classes of Boolean relations1with respect to the structure of the solution graphs of Boolean formulas built using these relations,which we call tight and non-tight relations.The solution graphs of formulas built from tight relations are characterized by certain structural properties.On the other hand we find non-tight sets of relations;formulas built from such sets of relations can express any solution graph.The boundary between these two classes differs from the boundary in Schaefer’s dichotomy.Schaefer showed that the satisfiability problem is solvable in polynomial time precisely for formulas built from Boolean relations all of which are bijunctive,or all of which are Horn,or all of which are dual Horn,or all of which are affine.The class of tight relations properly contains the classes of bijunctive,Horn,dual Horn,and affine relations.The main step in the proof of Schaefer’s dichotomy theorem is a result of independent interest known as Schaefer’s expressibility theorem.The crux of our results is a different expressibility theorem which we call the Faithful Expressibility Theorem.At a high level,this theorem asserts that given any Boolean relation with a solution graph,we can construct a formula using any non-tight set of relations,such that its solution graph is isomorphic to after certain adjacent vertices are merged.In addition to being an interesting structural result in its own right,the Faithful Expressibility Theorem implies that all non-tight relations have the samecomputational complexity for both the connectivity and the-connectivity problems.It also shows that thediameter of the solution graph of formulas obtainable such relations are polynomially related.As a consequence of the Faithful Expressibility Theorem we establish three dichotomy results.Thefirst isa dichotomy theorem for the-connectivity problem:Given a Boolean formula and two solutions andof,is there a path from to in?We show that-connectivity is solvable in linear time for formulas built from tight relations,and PSPACE-complete in all other cases.The second is a dichotomy theorem forthe connectivity problem:Given a Boolean formula,is connected?We show that connectivity is in coNP for formulas built from tight relations,and PSPACE-complete in all other cases.Finally,we establish a structural dichotomy theorem for the diameter of the connected components of the solution space of Boolean formulas.This result asserts that,in the PSPACE-complete cases,the diameter of the connected components can be exponential,but in all other cases it is linear.Technical Contributions.In Schaefer’s Dichotomy Theorem,NP-hardness of satisfiability was a consequence of an expressibility theorem,which asserted that every Boolean relation can be obtained as a projection over a formula built from clauses from any“hard”set of relations(i.e.a set in which at least one relation is not bijunctive,at least one is not Horn,at least one is not dual Horn,and at least one is not affine).Schaefer’s notion of expressibility is inadequate for our problem,so we introduce and work with a delicate and more strict notion of expressibility,which we call faithful expressibility.Intuitively,faithful expressibility means that,in addition to definability via a projection,the space of witnesses of the existential quantifiers in the projection has certain strong connectivity properties that allow us to capture the graph structure of the relation that is being defined.It should be noted that Schaefer’s Dichotomy Theorem can also be proved using a Galois connection and Post’s celebrated classification of the lattice of Boolean clones(see[4]).This method,however,does not appear to apply to connectivity,as the boundaries discovered here cut across Boolean clones.Thus,the use of faithful expressibility or some other refined definability technique seems unavoidable.The main technical challenge in this work is the proof of the Faithful Expressibility Theorem,which is proved via a series of reductions.To prove it,we identify the simplest non-tight relations:these are ternary relations whose graph is a path of length between assignments at Hamming distance.We show that one can faithfully express such a path from any non-tight set of relations.Next,we show that these paths can faithfully express all3-CNF clauses,which are then easily shown to faithfully express any relation.The Faithful Expressibility Theorem allows us to focus on a specific non-tight set of relations in order to establish the hard part of our dichotomies,We show that both connectivity and-connectivity are hard for 3-CNF formulas;this is proved by a reduction from a generic PSPACE computation.Similarly,we show that formulas built from non-tight relations can have large diameter by explicitly constructing a3-CNF formula on variables whose diameter is exponential in.Our upper bounds for tight sets of relations are proved using structural properties that characterize the solu-tion graphs.For tight sets of relation,we show that every component has a unique minimum element,or every component has a unique maximum element,or the Hamming distance coincides with the shortest-path distance in the relation.These properties are inherited by every formula built from a tight set of relations,and yield both small diameter and linear algorithms for-connectivity.An intriguing byproduct of our work is that we have identified a broad class of NP-complete satisfiability problems-those built from tight relations-that have simple structural properties,such as linear diameter.It would be interesting to investigate if these properties make random instances built from tight relations easier for WalkSAT and similar heuristics,and if so,whether such heuristics are amenable to rigorous analysis.Organization of this Paper.In Section2we introduce the main concepts precisely,and state our results.We prove the two sides of the dichotomy in Sections3and4respectively.Finally,we will discuss a few open questions and conjectures in Section5.An extended abstract of this paper appears in ICALP’06[10].2Basic Concepts and Statements of ResultsA logical relation is a non-empty subset of,for some;is the arity of.Let be afinite set of logical relations.A CNF-formula over a set of variables is afinite conjunction of clauses built using relations from,variables from,and the constants and;this means that each is an expression of the form,where is a relation of arity,and each is a variable in or one of the constants,.The satisfiability problem S AT associated with afinite set of logical relations asks:given a CNF-formula,is it satisfiable?All well known restrictions of Boolean satisfiability,such as-S AT,N OT-A LL-E QUAL-S AT,and P OSITIVE-IN-S AT,can be cast as S AT problems,for a suitable choice of.For instance,P OSITIVE1-IN-3S AT is S AT,where.Schaefer[20]identified the complexity of every satisfiability problem S AT.To state Schaefer’s main result,we need to define some basic concepts.Definition1Let be a logical relation.1.is bijunctive if it is the set of solutions of a2-CNF formula.2.is Horn if it is the set of solutions of a Horn formula,where a Horn formula is a CNF formula such thateach conjunct has at most one positive literal.3.is dual Horn if it is the set of solutions of a dual Horn formula,where a dual Horn formula is aCNF formula such that each conjunct has at most one negative literal.4.is affine if it is the set of solutions of a system of linear equations over.Each of these types of logical relations can be characterized in terms of closure properties[20].A relation is bijunctive if and only if it is closed under the majority operation(if,then, where is the vector whose-th bit is the majority of).A relation is Horn if and only if it is closed under(if,then,where,is the vector whose-th bit is).Similarly, is dual Horn if and only if it is closed under.Finally,is affine if and only if it is closed under. Definition2A set of logical relations is Schaefer if at least one of the following holds:1.Every relation in is bijunctive.2.Every relation in is Horn.3.Every relation in is dual Horn.4.Every relation in is affine.Theorem1(Schaefer’s Dichotomy Theorem[20])If is Schaefer,then S AT is in P;otherwise,S AT is NP-complete.Note that the closure properties of Schaefer sets yield a cubic algorithm for determining,given afinite set of relations,whether S AT is in P or NP-complete(the input size is the sum of the sizes of relations in).Here,we are interested in the connectivity properties of the space of solutions of CNF-formulas.If is a CNF-formula with variables,then denotes the subgraph of the-dimensional hypercube induced by the solutions of.Thus,the vertices of are the solutions of,and there is an edge between two solutions of precisely when they differ in a single variable.We consider the following two algorithmic problems for CNF-formulas.1.The connectivity problem C ONN:given a CNF-formula,is connected?2.The st-connectivity problem ST-C ONN:given a CNF-formula and two solutions and of,isthere a path from to in?To pinpoint the computational complexity of ST-C ONN and C ONN,we need to introduce certain new types of relations.Definition3Let be a logical relation.1.is componentwise bijunctive if every connected component of is bijunctive.2.is OR-free if the relation cannot be obtained from by setting of thecoordinates of to a constant.In other words,is OR-free if is not definable from byfixing variables.3.is NAND-free if is not definable from byfixing variables.The next lemma follows from the closure properties of bijunctive,Horn,and dual Horn relations. Lemma1Let be a logical relation.1.If is bijunctive,then is componentwise bijunctive.2.If is Horn,then is OR-free.3.If is dual Horn,then is NAND-free.4.If R is affine,then is componentwise bijunctive,OR-free,and NAND-free.These containments are proper.For instance,is componentwise bijunctive,but not bijunctive as.We are now ready to introduce the key concept of a tight set of relations.Definition4A set of logical relations is tight if at least one of the following three conditions holds:1.Every relation in is componentwise bijunctive;2.Every relation in is OR-free;3.Every relation in is NAND-free.In view of Lemma1,if is Schaefer,then it is tight.The converse,however,does not hold.It is also easy to see that there is a polynomial-time algorithm for testing whether a givenfinite set of logical relations is tight.The main step in the proof of Schaefer’s dichotomy theorem is a result known as Schaefer’s expressibility theorem.Similarly,the crux of our results is the following theorem which we will call the Faithful Expressibility Theorem.At a high level,this theorem asserts that for any Boolean relation with a solution graph,we can construct a formula using any non-tight set of relations,such that its solution graph is isomorphic to after certain adjacent vertices are merged.See section4for a precise definition of faithful expressibility. Theorem2(Faithful Expressibility Theorem)Let be a set of relations that is not tight.Every relation is faithfully expressible from.Using the Faithful Expressibility Theorem,we obtain dichotomy theorems for the computational complexity of C ONN and ST-C ONN.Theorem3Let be afinite set of logical relations.If is tight,then C ONN is in coNP;otherwise,it is PSPACE-complete.Theorem4Let be afinite set of logical relations.If is tight,then ST-C ONN is in P;otherwise,ST-C ONN is PSPACE-complete.We also show that if is tight,but not Schaefer,then C ONN is coNP-complete.The dichotomy in the computational complexity of C ONN and ST-C ONN is accompanied by a parallel structural dichotomy in the size of the diameter of(where,for a CNF-formula,the diameter ofis the maximum of the diameters of the components of).Theorem5Let be afinite set of logical relations.If is tight,then for every CNF-formula,the diameter of is linear in the number of variables of;otherwise,there are CNF-formulas such that the diameter of is exponential in the number of variables of.Our results and their comparison to Schaefer’s Dichotomy Theorem are summarized in the table below.ST-C ONN DiameterP coNPNP-complete coNP-completeNP-complete PSPACE-complete As an example,the set,where,is tight,but not Schaefer.It follows that S AT is NP-complete(recall that this problem is P OSITIVE1-IN-3S AT),ST-C ONN is in P,and C ONNis coNP-complete.Consider also the set,where.This set is not tight, hence S AT is NP-complete(this problem is P OSITIVE N OT-A LL-E QUAL3-S AT),while both ST-C ONN and C ONN are PSPACE-complete.We conjecture that if is Schaefer,then C ONN is in P.If this conjecture is true,it will follow that the complexity of C ONN exhibits a trichotomy:if is Schaefer,then C ONN is in P;if is tight,but not Schaefer,then C ONN is coNP-complete;if is not tight,then C ONN is PSPACE-complete.3The Easy Case of the Dichotomy:Tight Sets of RelationsIn this section,we explore some structural properties for the solution graphs of tight sets of relations.These properties provide simple algorithms for C ONN and ST-C ONN for tight sets,and also guarantee that for such sets,the diameter of of CNF-formula is linear.We will use to denote Boolean vectors,and and to denote vectors of variables.We writeto denote the Hamming weight(number of’s)of a Boolean vector.Given two Boolean vectors and,we write to denote the Hamming distance between and.Finally,if and are solutions of a Boolean formula and lie in the same component of,then we write to denote the shortest-path distance between and in.3.1Componentwise Bijunctive Sets of RelationsLemma2Let be a set of componentwise bijunctive relations and a CNF-formula.If and are two solutions of that lie in the same component of,then.P ROOF:Considerfirst the special case in which every relation in is bijunctive.In this case,is equivalent to a2-CNF formula and so the space of solutions of is closed under majority.We show that there is a path in from to,such that along the path only the assignments on variables with indices from the set change.This implies that the shortest path is of length by induction on.Consider any path in.We construct another path by replacing byfor,and removing repetitions.This is a path because for any and differ in at most one variable.Furthermore,agrees with and for every for which.Therefore,along this path only variables in areflipped.For the general case,we show that every component of is the solution space of a2-CNF formula .Let be the component of which contains and.Let be a relation with two components, each of which are bijunctive.Consider a clause in of the form.The projection of onto is itself connected and must satisfy.Hence it lies within one of the two components, assume it is.We replace by.Call this new formula.consists of all components of whose projection on lies in.We repeat this for every clause.Finally we are left with a formula over a set of bijunctive relations.Hence is bijunctive and is a component of .So the claim follows from the bijunctive case.Corollary1Let be a set of componentwise bijunctive relations.Then1.For every CNF with variables,the diameter of each component of is bounded by.2.ST-C ONN is in P.3.C ONN is in coNP.P ROOF:The bound on diameter is an immediate consequence of Lemma2.The following algorithm solves ST-C ONN given vertices.Start with.At each step,find a variable so that andflip it,until we reach.If at any stage no such variable exists,then declare that and are not connected.If the and are disconnected,the algorithm is bound to fail.So assume that they are connected.Correctness is proved by induction on.It is clear that the algorithm works when .Assume that the algorithm works for.If and are connected and are distance apart,Lemma2 implies there is a path of length between them in.In particular,the algorithm willfind a variable to flip.The resulting assignment is at distance from,so now we proceed by induction.Next we prove that C ONN coNP.A short certificate that the graph is not connected is a pair of assign-ments and which are solutions from different components.To verify that they are disconnected it suffices to run the algorithm for ST-C ONN.3.2OR-free and NAND-free Sets of RelationsWe consider sets of OR-free relations.Sets of NAND-free relations are handled dually.Define the coordinate-wise partial order on Boolean vectors as follows:if,for each.Lemma3Let be a set of OR-free relations and a CNF-formula.Every component of contains a minimum solution with respect to the coordinate-wise order;moreover,every solution is connected to the minimum solution in the same component via a monotone path.P ROOF:We call a satisfying assignment locally minimal,if it has no neighboring satisfying assignments that are smaller than it.We will show that there is exactly one such assignment in each component of.Suppose there are two distinct locally minimal assignments and in some component of.Consider the path between them where the maximum Hamming weight of assignments on the path is minimized.If there are many such paths,pick one where the smallest number of assignments have the maximum Hamming weight. Denote this path by.Let be an assignment of largest Hamming weight in the path.Then and,since and are locally minimal.The assignments and differ in exactly2variables,say,in and.So.Let be such that ,and for.If is a solution,then the pathcontradicts the way we chose the original path.Therefore,is not a solution.This means that there is a clause that is violated by it,but is satisfied by,,and.So the relation corresponding to that clause is not OR-free,which is a contradiction.The unique locally minimal solution in a component is its minimum solution,because starting from any other assignment in the component,it is possible to keep moving to neighbors that are smaller,and the only time it becomes impossible tofind such a neighbor is when the locally minimal solution is reached.Therefore,there is a monotone path from any satisfying assignment to the minimum in that component.Corollary2Let be a set of OR-free relations.Then1.For every CNF with variables,the diameter of each component of is bounded by.2.ST-C ONN is in P.3.C ONN is in coNP.P ROOF:Given solutions and in the same component of,there is a monotone path from each to the minimal solution in the component.This gives a path from to of length at most.To check if and are connected,we just check that the minimal assignments reached from and are the same.3.3The Complexity of C ONN for Tight Sets of RelationsWe can further specify the complexity of C ONN for the tight cases which are not Schaefer,using a result of Juban[12].Lemma4For tight,but not Schaefer,C ONN is coNP-complete.P ROOF:The problem A NOTHER-S AT is:given a formula in CNF and a solution,does there exist a solution?Juban([12],Theorem2)shows that if is not Schaefer,then A NOTHER-S AT is NP-complete. He also shows([12],Corollary1)that if is not Schaefer,then the relation is expressible from through substitutions.Since is not Schaefer,A NOTHER-S AT is NP-complete.Let be an instance of A NOTHER-S AT on variables.We define a CNF formula on asIt is easy to see that is connected if and only if is the unique solution to.Further we can show that C ONN is in P if is affine or bijunctive.Thus the only tight cases for which C ONN is not known to be coNP-complete or in P are Horn and dual-Horn.We conjecture that these problems are in P.01110(a)(b)(c)Figure1:Expressing the relation using N OT-A LL-E QUAL relations.(a)The graph of;(b)The graph of a faithful expression:.(c)The graph of an unfaithful expression:. In both cases,but only in thefirst case the connectivity is preserved.4The Hard Case of the Dichotomy:Non-Tight Sets of RelationsWe will show that all non-tight sets of relations lead to solution graphs that have identical properties in a natural sense that is captured in the notion of faithful expressibility.We define this notion in Section4.1,and prove the Faithful Expressibility Theorem in Section4.2.This theorem implies that the complexity of the connectivity questions for all such sets is the same,and the possible diameter of components of the solution graph is also related polynomially.In section4.3we will prove that for3-CNF formulas the connectivity questions are PSPACE-complete,and the diameter can be exponential.This fact together with the Faithful Expressibility Theorem implies the hard side of all of our dichotomy results.4.1Faithful ExpressibilityIn his dichotomy theorem,Schaefer[20]used the following notion of expressibility:a relation is expressible from a set of relations if there is a CNF-formula so that.This notion,is not sufficient for our purposes.Instead,we introduce a more delicate notion,which we call faithful expressibility. Intuitively,we view the relation as a subgraph of the hypercube,rather than just a subset,and require that this graph structure be also captured by the formula.Definition5A relation is faithfully expressible from a set of relations if there is a CNF-formula such that the following condition hold:1.;2.For every,the graph is connected;3.For with,there exists such that and are solutions of.For,the witnesses of are the’s such that is true.The last two conditions say that the witnesses of are connected,and that neighboring have a common witness.This allows us to simulate an edge in by a path in,and thus relate the connectivity properties of the solution spaces.There is however,a price to pay:it is much harder to come up with formulas that faithfully express arelation.An example is when is the set of all paths of length in,a set that plays a crucial role in our proof.While3-S AT relations are easily expressible from in Schaefer’s sense,the CNF-formulas that faithfully express3-S AT relations are fairly complicated and have a large witness space.An example of the difference between a faithful and an unfaithful expression is shown in Figure4.1. Lemma5Let and be sets of relations such that every is faithfully expressible from.Given a CNF-formula,one can efficiently construct a CNF-formula such that:1.;2.if are connected in by a path of length,then there is a path from to inof length at most;3.If are connected in,then for every witness of,and every witness of,there is apath from to in.P ROOF:Suppose is a formula on variables that consists of clauses.For clause,assume that the set of variables is,and that it involves relation.Thus,is.Let be the faithful expression for from,so that.Let be the vectorand let be the formula.Then.Statement follows from by projection of the path on the coordinates of.For statement,con-sider that are connected in via a path.For every, and clause,there exists an assignment to such that both and are solu-tions of,by condition of faithful expressibility.Thus and are both solutions of, where.Further,for every,the space of solutions of is the product space of the solutions of over.Since these are all connected by condition of faith-ful expressibility,is connected.The following describes a path from to in:.Here indicates a path in.Corollary3Suppose and are sets of relations such that every is faithfully expressible from.1.There are polynomial time reductions from C ONN to C ONN,and from ST-C ONN to ST-C ONN.2.Given a CNF-formula with clauses,one can efficiently construct a CNF-formulasuch that the length of is and the diameter of the solution space does not decrease.4.2The Faithful Expressibility TheoremIn this subsection,we prove the Faithful Expressibility Theorem.The main step in the proof is Lemma6which shows that if is not tight,then we can faithfully express the3-clause relations from the relations in.If ,then a-clause is a disjunction of variables or negated variables.For,let be the set of all satisfying truth assignments of the-clause whosefirst literals are negated,and let. Thus,CNF is the collection of-CNF formulas.Lemma6If set of relations is not tight,is faithfully expressible from.。

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Yi Lue-mail:luyi@Bo HuSchool of Mechanical Engineering,Yanshan University,Qinhuangdao,Hebei066004,China Unified Solving Jacobian/Hessian Matrices of Some Parallel Manipulators With n SPS Active Legs and a Passive Constrained LegSome parallel manipulators with n spherical joint-prismatic joint-spherical joint(SPS)-type active legs and a passive constrained leg possess a larger capability of load bearing and are simple in structure of the active leg.In this paper,a unified and simple approach is proposed for solving Jacobian/Hessian matrices and inverse/forward velocity and ac-celeration of this type of parallel manipulators.First,a general parallel manipulator with n SPS-type active legs and one passive constrained leg in various possible serial structure is synthesized,and some formulae for solving the poses of constrained force/ torque and active/constrained force matrix are derived.Second,the formulae for solving extension of active legs,the auxiliary velocity/acceleration equation are derived.Third, the formulae for solving inverse/forward velocity and acceleration and a Jacobian matrix without thefirst-order partial differentiation and a Hessian matrix without the second-order partial differentiation are derived.Finally,the procedure is applied to three par-allel manipulators with four andfive SPS-type active legs and one passive constrained leg in different serial structures and to illustrate.͓DOI:10.1115/1.2771572͔Keywords:parallel manipulator,constrained leg,kinematics,Jacobian matrix,Hessian matrix1IntroductionSome parallel manipulators with3–6degree of freedoms ͑DOFS͒have been used in many practical applications͓1,2͔. Among them,the parallel manipulators with nϽ6SPS-type active legs and one passive constrained leg in various serial structures have attracted more attention.This kind of parallel manipulator has a larger capability of load bearing and a simple structure of active leg͓3–6͔.In the aspect of kinematics,Tesar proposed a kinematic influence coefficient of mechanisms͓7,8͔.Huang et al. studied thefirst-second-order kinematic influence coefficient ma-trices which are later proved to be Jacobian/Hessian matrices͓2͔. Duffy and Rico et al.analyzed kinematics of the6-6R parallel manipulator and solved acceleration of open/closed spatial chains by means of screw theory͓9,10͔.Joshi and Tsai studied the Jaco-bian matrix for mechanisms with less than6DOF by using screw theory͓11͔.Kim and Ryu derived a homogeneous Jacobian matrix formulation by three end-effector points͓12͔.Fang and Huang solved velocity/acceleration of a3-RPS͑revolute joint-prismatic joint-spherical joint͒manipulator by using thefirst-/second-order kinematic influence coefficient matrices͓13͔.Canfield et al.ana-lyzed velocity of parallel manipulators by truss transformations ͓14͔.Lu et al.solved the velocity/acceleration and Jacobian ma-trix of some spatial parallel manipulators by using an analytic approach and a computer-aided design͑CAD͒variation geometry approach͓15–17͔.Gallardo-Alvarado et al.analyzed kinematics and singularity of a4-DOF parallel manipulator using screw theory͓18͔.Using the principle of virtual work,Tsai solved in-verse dynamics of use Stewart–Gough manipulator͓19͔.Others studied Jacobian matrix and singularity͓20,21͔.Since each of the items in Jacobian/Hessian matrices is the first-/second-order partial differentiation,this imposes difficulties on kinematics analysis of parallel manipulators.In addition,the forward pose equations of parallel manipulators generally are the implicit functions and have multisolutions.Therefore,to solve forward velocity/acceleration by means of thefirst-/second-order matrix approach is quite complex.This paper focuses on a unified and simple approach for deriv-ing Jacobian/Hessian matrices of the inverse/forward velocity andacceleration for parallel manipulator with nϽ6SPS-type active legs and one passive constrained leg in various serial structures. Two types of parallel manipulators with four,five SPS-type active legs and one passive constrained leg are presented to illustrate how to solve their Jacobian/Hessian matrices and velocity/ acceleration.2The Common Kinematics2.1A Parallel Manipulator With n SPS-Type Active Legs and One Constrained Leg in Various Serial Structure.Gener-ally,a parallel manipulator with n SPS active legs and one passiveconstrained leg r0includes a base B,a platform m,and n linearSPS active legs r i͑i=1,2,...,nϽ6͒with the linear actuators and one passive constrained leg r0in various serial structures͑see Fig.1͒.Let͕m͖be a coordinate o-xyzfixed on m at o;͕B͖be a coordinate O-XYZfixed on B at O;e i be the distance from b i to o; and E i be the distance from B i to O.Each of r i connects m at joint b i with B at joint B i,provides oneconstraint to m,and bears an active force F ai exerted on and alongr i.The constrained leg r0connects m at joint o with B at joint O,provides6-n constraints to m,and bears6-n constrained forces ortorques exerted on r0.Contributed by the Mechanisms and Robotics Committee of ASME for publica-tion in the J OURNAL OF M ECHANICAL D ESIGN.Manuscript received August5,2006;final manuscript received November15,2006.Review conducted by Qizheng Liao.Dofs of the parallel manipulators can be calculated by adopting a revised Kutzbach–Grubler equation as follows ͓1,2͔M =6͑q 0−q −1͒+͚i =1qm i −M͑1͒where q 0is the number of links;q is the number of joints;m i is the DOF of the i th joint;m 1=1for the prismatic joint P or revo-lute joint R ;m 2=2for the universal joint U or cylinder joint C ;m 2=3for the spherical joint S ;and M 0is the local redundant DOF,which have no influence on the motion of parallel mechanism.Corresponding to the number n of r i ,many possible types of constrained legs r 0in serial structure can be synthesized ͑see Table 1͒.By comparing them with each other,a prismatic joint-universal joint ͑PU ͒-type passive leg for n =3;a prismatic joint-spherical joint ͑PS ͒-type for n =4;and a universal joint-prismatic joint-universal joint ͑UPU ͒-type for n =5are considered to be bet-ter in structure because the interference among r i and r 0can be avoided easily.2.2The Geometric Constraints for Determining the Pose of Constrained Wrench.When ignoring the friction of all joints in a parallel manipulator,the workloads can be simplified as a wrench ͑F ,T ͒,which include the inertia wrench of platform and legs due to their masses,the damping wrench due to velocity,the gravity,and the external working wrench,where F is a central force and T is a central torque.The wrench ͑FT ͒is applied onto m at o ,and is variables versus time.͑FT ͒or its components ͑F X ,F Y ,F Z ,T X ,T Y ,T Z ͒are balanced by n active forces F ai ͑i =1,2...,n ͒and a constrained wrench ͑F p T p ͒exerted on r 0͑Fig.1͒.F p is composed of n 1constrained forces F pj ͑j =1,...,n 1͒;and T p is composed of n 2constrained torques T pk ͑k =1,...,n 2͒.Their numbers satisfy n +n 1+n 2=6,n =M .In a parallel manipulator with n SPS-type active legs and one constrained leg in various serial structures,the passive constrained leg may be composed of some multi-DOF joints,such as sphericaljoint S ,universal joint U ,cylindrical joint C ,prismatic joint P ,and revolute joint R .In order to determine the pose of ͑F p T p ͒,an equivalent constrained leg r e must be constructed by replacing S with three intersecting revolute joints,U with 2crossed revolute joints,and C with a prismatic joint and a revolute joint,respectively.During the movement of the parallel manipulator,since the con-strained wrench ͑F p T p ͒do not do any work,three geometric con-straints for determining the poses of the constrained wrench are obtained as follows:1.Let v re be a velocity along prismatic joint P in equivalent active leg r e ,there must be F pj v re =0,i.e.,F pj ЌP .Thus,each of the constrained forces must be perpendicular to all the prismatic joints in r e .2.Let R e be an unit vector of revolute joint R in r e ,and let ␳ÃF pj be a torque of F pj about R ,where there must be R e ·͑␳ÃF pj ͒=0.Thus,each of the constrained forces must intersect or be parallel with all the revolute joints in r e .3.Let ␻re be a rotation speed about R in r e ,where there must be T pk ·␻re =0,i.e.,T pk ЌR .Thus,each of the constrained torques must be perpendicular to all the revolute joints in r e .2.3The Active Force and Constrained Wrench.In order to solve active forces and constrained wrench,a set of wrench bal-ancing equations of the parallel manipulator to m at o are obtained as follows͚i =1nFai ␦i+͚j =1n 1Fpj f j=−F ,n +n 1+n 2=6͑2͚͒i =1nFai e iϫ␦i +͚j =1n 1d j ϫFpj f j+͚k =1n 2Tpk ␶k=−Twhere f j is the unit vector of F pj ;d j is the arm vector of F pj to o ;and ␶k is the unit vector of T pk .Equation ͑2͒is transformed asG ͓F a 1,...,F an ,F p 1,...,F pn 1,T p 1,...,T pn 2͔T =−͓F T ͔T͑3͒where G is defined as a 6ϫ6force matrix.Its formula is ex-pressed as follows G =ͫ␦1¯␦nf 1¯f n 10¯e 1ϫ␦1¯e n ϫ␦n d 1ϫf 1¯d n 1ϫf n 1␶1¯␶n 2ͬ6ϫ6͑4͒Fig.1A parallel manipulator with n SPS active legs:a con-strained legTable 1Various possible serial structure of constrained leg r o for parallel manipulators with n <6SPS-type active legs n Various possible serial structure of constrained legUP UR CP CR PC PU 3PRP PPR PRR PPP RC RU RRP RPR RPP RRR SP SR UPR URR UPP URP UU PRPR PPPR PRRR PS PUP PPU PUR PRU PRRP PPRR PRPP 4PPPP RRRP RS RPU RUP RRU RUR RRPR RPRR RPRP RPPP RRRR RPC RCP RRC RCR PCP PPC PCR PRC CPR CRR CPP CRP UUP UUR UCP UCR US UPU UPC URC URPR URRR UPRR URU CCP CCR CUP CUR CS CPC CRC CPU CRU CPRR CRPR CRPP 5CRRP CRRR CRRR SC SU SPP SPR SRR SRP PPS PPCR PPRC PPRU PPPU PPPC PRPU PCC PCRR PRPC RPPC RRPU RPS RRPC RRRRR RCRRRRCRRURRRRURRUPPRRUPThus,F ai ,F pj ,T pk can be solved as follows͓F a 1,...,F an ,F p 1,...,F pn 1,T p 1,...,T pn 2͔6ϫ1T =−G −1͓F T ͔T͑5͒2.4Inverse Displacement Kinematics.The position vectors B iB of joints B i on B in ͕B ͖,the position vectors b i m of joints b i onm in ͕m ͖,and position vectors b i Bof b i in ͕B ͖can be written as ͓1,2͔B i B =΄X BiY Bi Z Bi΅,b i m =΄x biy bi z bi΅,R m B=΄x l y l z lx m y m z m x n y n z n΅,o B =΄X oY oZ o΅b i B =R m B b i m+o B͑6a ͒where R B m is a rotational transformation matrix from ͕m ͖to ͕B ͖;͑X o ,Y o ,Z o ͒are three position components of m at o in ͕B ͖;o B is a position vector of point o in ͕B ͖;and ͑x l ,x m ,x n ,y l ,y m ,y n ,z l ,z m ,z n ͒are nine orientation parameters of m ,which have the following constrained equations ͓1,2͔x l 2+x m 2+x n 2=1,y l 2+y m 2+y n 2=1z l 2+z m 2+z n 2=1,x l y l +x m y m +x n y n =0͑6b ͒x l z l +x m z m +x n z n =0,z l y l +z m y m +z n y n =0The length of active legs r i ͑i =1,2,...,n ͒and the vectors r i of r ican be solved as follows ͓1,2͔r i =͉b i B −B i B ͉,r i =͓X b i −X Bi Y b i −Y Bi Z b i −Z Bi ͔T͑7a ͒Based on the structure constraints of various parallel manipula-tors,6-M structure constrained equations can be derived.FromEqs.͑6a ͒and ͑7a ͒and 6-M structure constrained equations,Eq.͑7a ͒can be simplified as a set of common equations for solving r i as followsr i =r i ͑␪1,␪2,...,␪n ͒i =1,2,...,n͑7b ͒where ␪i are n independent parameters among six pose parameters ͑X o ,Y o ,Z o ,␣,␤,␭͒,n Ͻ6.The unit vector ␦i of r i and the vector e i of the line e i can be solved as below␦i =΄␦ix␦iy ␦iz ΅=r i r i =1r i΄X b i −X Bi Y b i −Y Bi Z b i −Z Bi΅,e i =΄e ixe iy e iz΅=b i B −o B͑8͒Suppose there are two vectors ␩,␵and a skew-symmetric matrix ␩ˆ.They must satisfy following equations ͓1,2͔␩=΄␩x␩y ␩z΅,␵=΄␵x ␵y ␵z΅,␩ˆ=΄−␩z ␩y ␩z−␩x−␩y␩x΅␩ϫ␵=␩ˆ␵,␩ˆT =−␩ˆ͑9͒2.5Inverse Velocity/Acceleration Along the i th Active Leg.Let V =͓v ␻͔T be a generalizing velocity of m at o ;v i be a velocity of m at joint b i ;then there must be v i =v +␻Ãe i ͑i =1,2...,n ͒.A velocity v ri along r i is derived as␯ri =␯i ·␦i =͑␯+␻ϫe i ͒·␦i =␦i ·␯+͑e i ϫ␦i ͒·␻=J ri V͑10͒J ri =͓␦i T ͑e i ϫ␦i ͒T ͔1ϫ6By differentiation of Eq.͑10͒with respect to time,an acceleration a ri along the i th active leg is derived as follows a ri =J ri A +J ˙riV A =͓a ␧͔T ,a =͓a x a ␯a z ͔T ,␧=͓␧x ␧␯␧z ͔T ͑11͒where J ˙ri is a 1ϫ6differentiation matrix of J ri;and A is a general acceleration of m .A differentiation ␦˙i of unit vector ␦i with re-spect to time is derived as follows␦˙i =͑␯i −␦i ␯ri ͒/r i =͑␯+␻ϫe i −␦i ␯ri ͒/r i͑12͒From Eq.͑12͒,the second item in Eq.͑11͒is expanded as J ˙ri V =͓␦˙i T ͑e ˙i ϫ␦i +e i ϫ␦˙i ͒T ͔V =␦˙i ·␯+͑e˙i ϫ␦i ͒·␻+͑e i ϫ␦˙i ͒·␻=␦˙i ·␯i +͑e ˙i ϫ␦i ͒·␻=␦˙i·␯+͑e ˙i ϫ␦i ͒␻+͑␻ϫe i ͒·␦˙i =͓͑␯i −␦i ␯ri ͒·␯i +r i ͑␻ϫe iϫ␦i ͒·␻͔/r i =͓␯i 2−␯ri 2+r i ␻ϫ͑␻ϫe i ͒·␦i ͔/r i͑13a ͒Let E 3ϫ3be a unit matrix.By means of Eqs.͑9͒and ͑10͒,the firstitem in Eq.͑13͒is derived as follows␯i 2=͑␯+␻ϫe i ͒·͑␯+␻ϫe i ͒=͑␯−e i ϫ␻͒·͑␯−e i ϫ␻͒=͑␯−eˆi ␻͒·͑␯−e ˆi ␻͒=͓͑E 3ϫ3−e ˆi ͔V ͒·͓͑E 3ϫ3−e ˆi ͔V ͒=͓͑E 3ϫ3−eˆi ͔V ͒T ͓͑E 3ϫ3−e ˆi ͔V ͒=V T ͫE 3ϫ3−e ˆi eˆi −eˆi 2ͬV ͑13b ͒From Eq.͑9͒,the second item in Eq.͑13a ͒is derived as follows␯ri 2=ͩV Tͫ␦ie i ϫ␦i͓ͬͪ͑␦i T ͑e i ϫ␦i ͒T ͔V ͒=VTͫ␦i ␦i T␦i ͑e i ϫ␦i ͒T ͑e i ϫ␦i ͒␦i T ͑e i ϫ␦͒͑e i ϫ␦i ͒TͬV ͑13c ͒From Eq.͑9͒,the third item in Eq.͑13a ͒is derived as follows ␻ϫ͑␻ϫe i ͒·␦i =͑␻ϫe i ͒·͑␦i ϫ␻͒=−e ˆi ␻·␦ˆi ␻=␻T e ˆi ␦ˆi␻=V Tͫ03ϫ303ϫ303ϫ3͑e ˆi ␦ˆi ͒3ϫ3ͬV ͑13d ͒From Eqs.͑13a ͒–͑13d ͒,Eq.͑11͒is transformed as followsa ri =J ri A +V Th ri V ,h ri =1r i΄i h 11i h 12¯i h 16ih 21i h 22¯ih 26]]]]i h 61ih 62¯ih 66΅6ϫ6͑14͒where h ri is a 6ϫ6sub-Hessian matrix.Since each of h ri only includes simple algebraic equations without any second-order dif-ferentiation,and a Hessian matrix is simplified obviously.All items in h ri ͑i =1,2,...,n ͒are derived ͑see the Appendix ͒.2.6Inverse Velocity/Acceleration.When a parallel manipu-lator has n SPS-type active legs r i ͑i =1,2,...,n Ͻ6͒and one constrained leg,from Eqs.͑10͒and ͑14͒,its general inverse velocity/acceleration v r /a r along r i can be solved as follows␯r =J r V ,a r =J r A +V T H r VJ r =΄␦1T ͑e 1ϫ␦1͒T]␦n T ͑e n ϫ␦n ͒T ΅n ϫ6H r =΄h 1]h n΅,␯r =΄␯r 1]␯rn΅,a r =΄a r 1]a rn΅͑15͒where J r is a n ϫ6Jacobian matrix;and H r is a Hessian matrix with n layers of 6ϫ6submatrices.Since n Ͻ6and J r is not asquare matrix,neither an inverse matrix J r −1nor forward velocity/acceleration can be derived.2.7The Auxiliary Velocity/Acceleration Equations.In or-der to solve the forward velocity/acceleration of parallel manipu-lators with n Ͻ6SPS-type active legs,and one constrained leg,a common 6ϫ6Jacobian matrix and a common Hessian matrix must be created.Since the constrained wrench ͑F p T p ͒do not do any work dur-ing the movement of parallel manipulator,there must beF pj f j ·␯+͑d j ϫF pj f j ͒·␻=0,͑j =1,¯n 1͒͑16͒T pk ␶k ·␻=0,͑k =1,¯n 2,n +n 1+n 2=6͒When removing F pj and T k from Eq.͑16͒,an auxiliary velocity matrix equation is derived as followsJ a V =0,J a =΄f 1T ͑d 1ϫf 1͒T]]f n 1T ͑d n 1ϫf n 1͒T01ϫ3␶1T ]]01ϫ3␶n 2T ΅͑6−n ͒ϫ6͑17͒where J a is a ͑6-n ͒ϫ6auxiliary Jacobian matrix.By differentia-tion Eq.͑17͒with respect to time,an auxiliary acceleration matrix equation is derived as follows,0=J a A +J ˙a V J ˙a=΄f ˙1T ͑d ˙1ϫf 1+d 1ϫf ˙1͒T ]]f ˙n 1T ͑d ˙n 1ϫf n 1+d n 1ϫf ˙n 1͒T 01ϫ3␶˙1T ]]01ϫ3␶˙n 2T ΅͑6−n ͒ϫ6=V T H a͑18͒where H a is an auxiliary Hessian matrix.2.8The Common Velocity and Acceleration Equations.By combing Eq.͑15͒with Eqs.͑17͒and ͑18͒,respectively,the com-mon inverse velocity and acceleration ͑v c a c ͒of this type of par-allel manipulators are derived as follows␯c =J V ,a c =J A +V T H V ␯c =ͫ␯r 0ͬ6ϫ1,a c =ͫa r 0ͬ6ϫ1,J =ͫJ r J aͬ6ϫ6,H =ͫH r H aͬ͑19͒where J is a common 6ϫ6Jacobian matrix;and H is a common Hessian matrix with six layers of 6ϫ6submatrices.From Eqs.͑4͒and ͑19͒,we obtain G =J T or J =G T .Thus,J can be used to solve the active forces and constrained wrench of this type of parallel manipulators.From Eq.͑19͒,the common forward velocity/acceleration of this type of parallel manipulator is derived as followsV =J −1␯c ,A =J −1͑a c −V TH V ͒͑20͒In Eqs.͑14͒,͑15͒,and ͑19͒,each item in J ,J −1,H r and its sub-matrix h ri does not include any first-/second-order partial differ-entiation.Therefore,the formulae for solving inverse/forward ve-locity and acceleration of parallel manipulators are obviously simplified.The inverse/forward velocity and acceleration can be solved in the following cases:When given the forward velocity V and the acceleration A of m ,from Eq.͑15͒,the inverse velocity v r and acceleration a r can be solved.When given the common inverse velocity v c and the common acceleration a c ,from Eq.͑20͒,the forward velocity V and accel-eration A can be solved.When solving A ,a differentiation matrix of auxiliary Jacobian matrix J a must be derived.Based on thepose of constrained wrench,each differentiation item in J ˙acan be derived.3A 4-SPS/PS Parallel ManipulatorA 4-SPS/PS parallel manipulator is composed of a platform m ,a baseB ,four SPS-type active legs r i ͑i =1,2,3,4͒with the linear actuators,and one PS-type passive constrained leg r o ͑see Fig.2͒.In order to avoid singularity,let m be a rectangle link with a long side l 1,a short side l 2,four vertices b i ,and a central point o .Let B be a square with sides L i =L ,four vertices B i ,and a central point O .Each of r i connects m with B by a spherical joint S on m at b i ,a prismatic joint P along r i ,and a spherical joint S on B at B i .r o connects m with B by a prismatic joint P fixed on m at o ,a spherical joint S attached to B at O ,and there is a structure con-straint of r o Ќm .In the 4-SPS/PS parallel manipulator,the number of links is q 0=11for one platform,four cylinders,four piston rods,and one base;the number of joints is q =14for 5P ,and 9S ;and the redundancy DOF is M 0=4for four SPS-type active legs rotating about their own axes.Thus,the DOF of the 4-SPS/PS parallel manipulator is calculated as followsM =6͑q 0−q −1͒+͚i =1qm i −M=6ϫ͑11−14−1͒+͑5ϫ1+9ϫ3͒−4=43.1Inverse Kinematics.B i B of B and b i mof m ͑i =1,2,3,4͒are derived as followsB 1B =12΄L −L 0΅,B 2B =12΄L L 0΅,B 3B =12΄−L L 0΅,B 4B =12΄−L −L 0΅Fig.2A 4-SPS/PS parallel manipulator and its force situationb 1m =12΄l 1−l 20΅,b 2m =12΄l 1l 20΅,b 3m =12΄−l 1l 20΅,b 4m =12΄−l 1−l 20΅͑21͒From Eqs.͑6a ͒and ͑21͒,b i B of m are derived as followsb 1B =12΄x l l 1−y l l 2+2X ox m l 1−y m l 2+2Y o x n l 1−y n l 2+2Z o ΅,b 2B =12΄x l l 1+y l l 2+2X o x m l 1+y m l 2+2Y o x n l 1+y n l 2+2Z o΅b 3B =12΄−x l l 1+y l l 2+2X o −x m l 1+y m l 2+2Y o −x n l 1+y n l 2+2Z o΅,b 4B =12΄−x l l 1−y l l 2+2X o−x m l 1−y m l 2+2Y o −x n l 1−y n l 2+2Z o΅͑22͒When r o Ќm ,there is o B=r o z .Thus by means of Eq.͑6b ͒yieldX o /z l =Y o /z m =Z o /z n =r o ,x l 2=1−y l 2−͑X o /r o ͒2y m 2=1−x m 2−͑Y o /r o ͒2͑23͒Let R B m be defined by three Euler rotations of ͑Z ,Y 1,Z 2͒,namely,a rotation of ␣about Z axis,followed by a rotation of ␤about Y 1axis,and a rotation of ␭about Z 2axis,where Y 1is formed by Y rotating about Z by ␣,and Z 2is formed by Z 1rotating about Y 1by␭.Thus,R B m and o Bare derived as followsR m B =΄c ␣·c ␤·c ␭−s ␣·s ␭−c ␣·c ␤·s ␭−s ␣·c ␭c ␣·s ␤s ␣·c ␤·c ␭+c ␣·s ␭−s ␣·c ␤·s ␭+c ␣·c ␭s ␣·s ␤−s ␤·c ␭s ␤·s ␭c ␤΅,o B =r o ΄c ␣·s ␤s ␣·s ␤c ␤΅͑24͒Comparing each item in R B m and oBin Eqs.͑6͒and ͑24͒,͑x l ,y l ,z l ,x m ,y m ,z m ,x n ,y n ,z n ͒and ͑X o ,Y o ,Z o ͒can be expressed by ͑␣,␤,␭,r o ͒.From Eqs.͑6b ͒,͑7a ͒,and ͑21͒–͑24͒,r i are derived as followsr 12=͑2L 2+l 12+l 22͒/4+r o 2−͓c ␤͑c ␣−s ␣͒͑l 1c ␭+l 2s ␭͒−͑s ␣+c ␣͒ϫ͑l 1s ␭−l 2c ␭͒+2r o s ␤͑c ␣−s ␣͔͒L /2r 22=͑2L 2+l 12+l 22͒/4+r o 2−͓c ␤͑c ␣+s ␣͒͑l 1c ␭−l 2s ␭͒−͑s ␣−c ␣͒ϫ͑l 1s ␭+l 2c ␭͒+2r o s ␤͑c ␣+s ␣͔͒L /2r 32=͑2L 2+l 12+l 22͒/4+r o2−͓c ␤͑c ␣−s ␣͒͑l 1c ␭+l 2s ␭͒−͑s ␣+c ␣͒ϫ͑l 1s ␭−l 2c ␭͒−2r o s ␤͑c ␣−s ␣͔͒L /2r 42=͑2L 2+l 12+l 22͒/4+r o2−͓c ␤͑c ␣+s ␣͒͑l 1c ␭−l 2s ␭͒−͑s ␣−c ␣͒ϫ͑l 1s ␭+l 2c ␭͒−2r o s ␤͑c ␣+s ␣͔͒L /2͑25͒From Eqs.͑8͒and ͑21͒–͑25͒,␦i and e i ͑i =1,2,3,4͒are derived asfollows␦1=12r 1΄x l l 1−y l l 2+2X o −L x m l 1−y m l 2+2Y o +L x n l 1−y n l 2+2Z o΅␦2=12r 2΄x l l 1+y l l 2+2X o −Lx m l 1+y m l 2+2Y o −L x n l 1+y n l 2+2Z o΅␦3=12r 3΄−x l l 1+y l l 2+2X o +L −x m l 1+y m l 2+2Y o −L −x n l 1+y n l 2+2Z o ΅␦4=12r 4΄−x l l 1−y l l 2+2X o +L −x m l 1−y m l 2+2Y o +L −x n l 1−y n l 2+2Z o ΅e 1=12΄x l l 1−y l l 2x m l 1−y m l 2x n l 1−y n l 2΅,e 2=12΄x l l 1+y l l 2x m l 1+y m l 2x n l 1+y n l 2΅e 3=12΄−x l l 1+y l l 2−x m l 1+y m l 2−x n l 1+y n l 2΅,e 4=12΄−x l l 1−y l l 2−x m l 1−y m l 2−x n l 1−y n l 2΅͑26͒From Eqs.͑21͒–͑26͒,␦i and e i which are expressed by four poseparameters ͑␣,␤,␭,r o ͒can be derived.In the PS-type passive constrained leg r o ,a spherical joint S can be replaced by three intersecting revolute joints ͑R 1,R 2,R 3͒.Thus,an equivalent constrained leg includes three intersecting revolute joints ͑R 1,R 2,R 3͒and one prismatic joint P .Their unit vectors are determined as followsR 1=΄001΅,R 2=΄−s ␣c ␣0΅,R 3=΄c ␣·s ␤s ␣·s ␤c ␤΅,P =o B r o=΄c ␣·s ␤s ␣·s ␤c ␤΅͑27͒3.2The Common Jacobian/Hessian Matrices.Based on thegeometric constraints of the constrained wrench in Sec.2.2,in the equivalent passive constrained leg,there must be ͑F p 1Ќo B ,F p 2Ќo B ,and F p 1,and F p 2intersecting or parallel with R 1,R 2,R 3͒.Thus,both F p 1and F p 2are exerted on r o at O .By means of Eq.͑9͒,their unit vectors f 1and f 2and differentiations are deter-mined as followsf 1=x =͓x l x m x n ͔T ,f 2=y =͓y l y m y n ͔T ͑28a ͒f ˙1=x ˙=␻ϫx =␻x ˆ,f ˙2=y ˙=␻ϫy =␻y ˆThe vectors ͑d 1,d 2͒of the distances from o to F p 1and F p 2andtheir differentiations are determined as follows d 1=d 2=−o B =−r o ͓s ␤−s ␣·c ␤c ␣·c ␤͔T ,d ˙1=d ˙2=−␯͑28b ͒From Eqs.͑17͒,͑18͒,͑28a ͒,and ͑28b ͒,the auxiliary Jacobian/Hessian matrices are derived as followsJ a =ͫf 1T ͑d 1ϫf 1͒Tf 2T ͑d 2ϫf 2͒T ͬ2ϫ6=ͫx −o B ϫx y −o B ϫyͬ2ϫ6J ˙a =ͫf ˙1T ͑d ˙1ϫf 1+d 1ϫf ˙1͒T f ˙2T ͑d ˙2ϫf 2+d 2ϫf ˙2͒T ͬ=ͫ␻xˆ−␯x ˆ−␻o ˆB x ˆ␻yˆ−␯y ˆ−␻o ˆB y ˆͬ=V Tͫh a 1h a 2ͬ͑29͒where h ai ͑i =1,2͒are two auxiliary sub-Hessian matrices.Bymeans of Eq.͑9͒,h ai are derived as followsh a 1=ͫ0−xˆxˆ−o ˆB x ˆͬ6ϫ6,h a 2=ͫ0−yˆyˆ−o ˆB y ˆͬ6ϫ6From Eqs.͑19͒and ͑29͒,the common inverse velocity v c andcommon inverse acceleration a c of the 4-SPS/PS parallel manipu-lator ͒are derived as follows␯c =J V ,a c =J A +V T H VJ =G T =ͫJ r J aͬ=΄␦1T ͑e 1ϫ␦1͒T ␦2T ͑e 2ϫ␦2͒T ␦3T ͑e 3ϫ␦3͒T ␦4T͑e 4ϫ␦4͒Tf 1T ͑d 1ϫf 1͒T f 2T ͑d 2ϫe 2͒T΅6ϫ6,H =ͫH r H aͬ=΄h r 1h r 2h r 3h r 4h a 1h a 2΅͑30͒where J is a common Jacobian matrix and H is a common Hes-sian matrix.4A 5-SPS/UPU Parallel ManipulatorA 5-SPS/UPU parallel manipulator is composed of a platformm ,a base B ,five SPS-type active legs r i ͑i =1,2,...,5͒with the linear actuators,and one UPU-type passive constrained leg r o ͑see Fig.3͒.In order to avoid singularity,let m be a pentangle with five different sides l i ,five vertices b i ,and a center point o ;let the distance from b i to o be e i =e ;let b 2and b 4lie in the x axis and b 3in the y axis;let the angle between e 1and y be 2␪,and the same for the angle between e 5and y .Let B be an equilateral pentangle with five sides L i =L ,five vertices B i ,and a center point O ;let the distance from A i to O be E i =E ;let B 3lie in Y ;let the angle between E 1and Y be 2␪and the same for the angle between E 5and Y ;and let the angle between E 2and X be ␪and the same for the angle between E 4and X .Each of r i connects m with B by a sphere joint S on m at b i ,a prismatic joint P ,and a sphere joint S on B at B i .The UPU-type passive leg r o connects m with B by a universal joint U m on m at o ,a P along r o ,and a universal joint U B on B at O .U B can be replaced by two cross revolute joints R B 1and R B 2.U m can be replaced by two cross revolute joints R m 1and R m 2.In addition,there are some structure constraints of ͑R B 1ЌR B 2,R m 1ЌR m 2,with R B 1being coincident with X ,R m 1being coincident with z ,R B 2ʈR m 2,R m 2Ќr o ,and R B 2Ќr o ͒.From these structure constraints,it can be verified that z intersects with the X axis at point c ͑i.e.,X ,r o ,and z lie in a plane ⌬Ooc ͒.In the 5-SPS/UPU parallel manipulator,the number of links is q 0=14for one platform,six cylinders,six piston rods,and one base;the number of joints is q =18for 6P ,2U ,and 10S ;and the redundancy DOFs is M 0=5for five SPS-type active legs rotating about their own axes.Thus,the DOF of the 5-SPS/UPU parallel manipulator is M =6͑q 0−q −1͒+͚i =1qm i −M=6ϫ͑14−18−1͒+͑6ϫ1+2ϫ2+10ϫ3͒−5=54.1Inverse Kinematics.B i B of B and b i mof m ͑i =1,...,5͒are derived as followsB 1B =E ΄s 2␪−c 2␪0΅,B 2B =E ΄c ␪s ␪0΅,B 3B =΄0E΅B 4B =E ΄−c ␪s ␪΅,B 5B =E ΄−s 2␪−c 2␪΅͑31͒b 1m =e ΄s 2␪−c 2␪0΅,b 2m =΄e00΅,b 3m =΄0e 0΅,b 4m =΄−e 00΅b 5m =e ΄−s 2␪−c 2␪0΅,␪=18degFrom Eqs.͑6a ͒and ͑31͒,b i B of m ͑i =1,2,...,5͒are derived as followsb 1B =΄x l es 2␪−y l ec 2␪+X ox m es 2␪−y m ec 2␪+Y o x n es 2␪−y n ec 2␪+Z o΅,b 2B =΄x l e +X ox m e +Y ox n e +Z o΅b 3B =΄ey l +X oey m +Y oey n +Z o΅͑32͒b 4B =΄−x l e +X o−x m e +Y o −x n e +Z o΅,b 5B =΄−x l es 2␪−y l ec 2␪+X o−x m es 2␪−y m ec 2␪+Y o−x n es 2␪−y n ec 2␪+Z o΅Since vectors ͑o B ,z ,X ͒lie in a plane ⌬Ooc ,there must be a con-strained equation as followsΈX o Y o Z oz l z m z n 100Έ=0,i.e.,Y o =z m Z o z n͑33͒Let R mBof the 5-SPS/UPU parallel manipulator be the same as that of the 4-SPS/PS parallel manipulator.From Eqs.͑24͒and ͑33͒,o B is derived as followso B =͓X o Z o s ␣·tan ␤Z o ͔T͑34͒From Eqs.͑6b ͒,͑7a ͒,and ͑31͒–͑34͒,r i arederivedFig.3A 5-SPS/UPU parallel manipulator and its forcesituation。

普通话水平测试用普通话词语表（一）1 阿ü 5 矮ǎi9爱护àihù 13 安定ündìng 17 安全ünquán 21 安装ünzhuüng 25 按àn 29 暗àn 33 熬áo 37 八bü 41 把bǎ45 爸爸bàbà 49 白色báisa 53 百姓bǎixìng 57 败bài 61 颁布bünbù65 板bǎn 69 办bàn73 办事bànshì 77 半径bànjìng 81 伴bàn 85 帮büng 89 棒bàng93 包干儿büogànr 97 包装büozhuüng 101 饱bǎo105 宝贵bǎoguì 109 保存bǎocún 113 保守bǎoshǒu 117 保证bǎozhang 121 报复bào?fù2 阿姨üyí 6爱ài10 爱情àiqíng 14 安静ünjìng 18 安慰ün wai 22 氨ün26 按照ànzhào 30 暗示ànshì 34 敖áo 38 巴bü 42 把握bǎw? 46 罢bà 50 白天bái?tiün54 摆bǎi 58 拜bài 62 搬bün66 板凳bǎndang 70 办法bànfǎ 74 半bàn78 半天bàntiün 82 伴随bànsuí 86 帮忙büngmáng 90 傍晚bàngwǎn 94 包含büohán 98 孢子büozǐ 102 饱和bǎoh? 106 宝石bǎoshí 110 保管bǎoguǎn 114 保卫bǎowai 118 报bào122 报告bàogào3 挨üi 7爱国àigu? 11 爱人àiren 15 安排ünpái 19 安心ünx?n 23氨基酸ünj?suün 27 案àn31 暗中ànzhōng 35 奥秘àomì 39 扒bü43 把儿bàr 47 罢工bàgōng 51 百bǎi 55 摆动bǎid?ng 59 班bün63 搬家bünjiü 67 版块bǎnkuài 71办公室bàngōngshì75 半导体bàndǎotǐ79 半夜bànya 83 伴奏bànz?u 87 帮助büngzhù 91 包büo 95 包括büoku?99 炮püo 103 宝bǎo 107 保bǎo111 保护bǎohù 115 保险bǎoxiǎn 119 报酬bào?ch?u 123 报刊bàokün4 挨ái8爱好àihào12 安ün16 安培ünp?i 20 安置ünzhì 24 岸àn28 案件ànjiàn 32 凹üo36 奥运会àoyùnhuì40 拔bá 44 爸bà 48 白bái52 百年bǎinián 56 摆脱bǎituō 60 般bün 64 搬运bünyùn68 版bǎn72 办理bànlǐ 76 半岛bàndǎo 80 扮演bànyǎn 84 瓣bàn88 榜样bǎngyàng92 包袱büofu 96 包围büow?i 100 薄báo104 宝贝bǎobai 108 保持bǎochí 112 保留bǎoliú 116 保障bǎozhàng120 报道bàodào 124 报名bàomíng125 报纸bàozhǐ 129 暴露bàolù 133 杯 byi 137 悲剧byijù141 备 bai 145 倍 bai 149 被子 baizi 153 本btn157 本能 btnn?ng 161 本事 btnshi 165 苯 btn 169 蹦bang173 鼻子bízi 177 比例bǐlì 181 比重bǐzh?ng 185 笔记bǐjì 189 必然bìrán 193 必要bìyào 197 闭合bìh? 201 避免bìmiǎn 205 边界biünjia 209 编biün 213 鞭biün217 变动biànd?ng 221 变化biànhuà 225 变态biàntài 229 便利biànlì 233 辨别biànbi? 237 辩证法biànzhangfǎ241 标语biüoyǔ 245 表biǎo249 表明biǎomíng126 抱bào 130 暴雨bàoyǔ 134 背 byi 138 北 bti 142 背 bai 146 被 bai 150 辈bai154 本地btndì 158 本人 btnr?n 162 本体btntǐ 166 奔 ban 170 逼 b? 174 比bǐ 178 比如bǐrú 182 彼bǐ186 笔者bǐzht 190必然性bìránxìng194 毕竟bìjìng 198 壁bì 202 臂bì206 边境biünjìng 210 编辑biünjí 214 鞭子biünzi 218 变法biànfǎ 222 变换biànhuàn 226 变形biànxíng 230 便于biànyú 234 辨认biànran238 标biüo242 标志biüozhì 246 表层biǎoc?ng 250 表皮biǎopí127 暴动bàod?ng 131 爆发bàofü 135 悲哀byiüi 139 北方btifüng 143 背后baih?u 147 被动 baid?ng 151 奔byn155 本来btnlái 159 本身 btnshyn 163 本性btnxìng 167 笨 ban 171 鼻bí175 比价bǐjià 179 比赛bǐsài 183 彼此bǐcǐ 187 必bì191 必须bìxū 195 毕业bìya 199 壁画bìhuà 203 边biün207 边区biünqū 211 编写biünxit 215 扁biǎn219 变革biàng? 223 变量biànliàng227 变异biànyì 231 遍biàn235 辩护biànhù 239 标本biüobtn 243 标准biüozhǔn 247 表达biǎodá 251 表情biǎoqíng128 暴力bàolì 132 爆炸bàozhà 136 悲惨byicǎn 140 贝bai144 背景baijǐng 148 被告baigào 152 奔跑bynpǎo 156 本领btnlǐng 160 本事btnshì 164 本质btnzhì 168 崩溃byngkuì 172 鼻孔bíkǒng 176 比较bǐjiào 180 比喻bǐyù 184 笔bǐ188 必定bìdìng 192 必需bìxū 196 闭bì 200 避bì204 边疆biünjiüng 208 边缘biünyuán 212 编制biünzhì 216 变biàn220 变更biàngyng 224 变迁biànqiün 228 便biàn 232 辨biàn236 辩证biànzhang 240 标题biüotí 244 标准化biüozhǔnhuà248 表面biǎomiàn 252 表示biǎoshì253 表述biǎoshù 257 表扬biǎoyáng 261 别 265 兵 269 饼bia b?ng bǐng254 表现biǎoxiàn 258 表彰biǎozhüng 262 宾 270 屏 274 病b?n pǐng bìng266 兵力b?nglì255 表象biǎoxiàng 259 别 263 冰 267 丙 271 并bi? b?ng bǐng bìng256 表演biǎoyǎn 260 别人 bi??r?n 264 冰川b?ngchuün 268 柄bǐng272 并且bìngqit 276 病毒bìngdú 280 拨bō273 并用bìngy?ng 277 病理bìnglǐ 275 病变bìngbiàn 279 病人bìngr?n 278 病情bìngqíng 281 波285 玻璃 289 播种 293 搏斗 297 补 301 捕 305 不309 不曾 313 不等 317 不妨 321 不管 325 不及 329 不堪 333 不良 337 不免341 不容 345 不想 349 不要 353 不止 357 布置 361 步子 365 部落 369 擦 373 材377 财富 381 采bōbō?lí bōzh?ng b?d?u bǔ bǔ bùbùc?ng bùdtng bùfáng bùguǎn bùjí bùkün bùliáng bùmiǎn bùr?ng bùxiǎng bùyào bùzhǐ bùzhì bùzi bùlu? cü cái cáifù cǎi282 波长bōcháng 286 剥夺bōdu? 290 伯 b? 294 薄b?298 补偿bǔcháng 302 捕捞bǔlüo 306 不安bù’ün 310 不错bùcu? 314 不定bùdìng 318 不服bùfú 322 不光bùguüng 326 不禁bùj?n 330 不可bùkt 334 不料bùliào 338 不怕bùpà 342 不如bùrú 346 不行bùxíng 350 不宜bùyí 354 不足bùzú 358 步bù 362 部bù 366 部门bùm?n 370 猜cüi374 材料cáiliào 378 财力cáilì 382 采访cǎifǎng283 波动bōd?ng 287 剥削bōxuy 291 脖子 b?zi 295 薄弱 b?ru? 299 补充bǔchōng 303 捕食bǔshí 307 不必bùbì 311 不但bùdàn 315 不断bùduàn 319 不够bùg?u 323 不过bùgu? 327 不仅bùjǐn 331 不快bùkuài 335 不论bùlùn 339 不平bùpíng 343 不时bùshí 347 不幸bùxìng 351 不已bùyǐ 355 布bù359 步伐bùfá 363 部队bùduì 367 部署bùshǔ 371 才cái 375 财cái379 财务cáiwù 383 采购cǎig?u284 波浪 288 播种 292 博士 296 薄300 补贴 304 捕捉 308 不便 312 不当 316 不对 320 不顾 324 不合 328 不久332 不利 336 不满 340 不然 344 不惜 348 不许 352 不用 356 布局 360 步骤 364 部分 368 部位 372 才能 376 财产 380 财政 384 采集bōlàng bōzhǒng b?shì b?bǔtiy bǔzhuō bùbiàn bùdüng bùduì bùgù bùh? bùjiǔ bùlì bùmǎn bùrán bùx?bùxǔ bùy?ng bùjú bùzh?u bùfen bùwai cáin?ng cáichǎn cáizhang cǎijí385 采取cǎiqǔ 389 踩cǎi393 参观cünguün 397参数cünshù 401 残酷cánkù 405 仓cüng 409 舱cüng413 操作cüozu? 417 草案cǎo’àn 421 侧 ca 425 测定cadìng 429 层 c?ng 433 叉chü437 差距chüjù 441 茶馆儿cháguǎnr 445 叉chǎ 449 拆chüi 453 产chǎn457 产生chǎnshyng 461 阐明chǎnmíng 465 长城chángch?ng469 长久chángjiǔ 473 场cháng 477 常cháng481 常数chángshù 485 场地chǎngdì 489 唱chàng 493 超额chüo’? 497 朝cháo501 潮湿cháosh? 505 车间chyjiün 509 车子 chyzi386 采用cǎiy?ng 390 菜cài394 参加cünjiü 398 参与cünyù 402 残余cányú 406 仓库cüngkù 410 藏cáng 414 曹cáo418 草地cǎodì 422 侧面camiàn 426 测量caliáng 430 层次c?ngcì 434 差chü438 差异chüyì 442 茶叶cháya 446 差chà 450 差chà454 产地chǎndì 458 产物chǎnwù 462 阐述chǎnshù 466 长处cháng?chù 470 长期chángq? 474 肠cháng 478 常规chánggu? 482 厂chǎng486 场合chǎngh? 490 抄chüo494 超过chüogu? 498 朝廷cháotíng 502 吵chǎo 506 车辆chyliàng510 扯cht387 彩cǎi 391 蔡cài395 参考cünkǎo 399 参照cünzhào 403 蚕cán 407 苍白cüngbái411 操cüo 415 槽cáo419 草原cǎoyuán 423 侧重 cazh?ng 427 测验cayàn 431 曾c?ng435 差别chübi? 439 插chü 443 查chá447差不多chà?bùduō 451 柴chái455 产量chǎnliàng 459 产业chǎnya 463 颤抖chàndǒu 467 长度chángdù 471 长远chángyuǎn 475 尝cháng479 常年chángnián 483 厂房chǎngfáng 487 场面chǎngmiàn 491 超chüo 495 超越chüoyua499 潮cháo 503 炒chǎo507 车厢chyxiüng 511 彻底chadǐ388 彩色cǎi sa 392 参cün 396 参谋cünm?u 400 残cán404 灿烂cànlàn 408 苍蝇cüngying 412 操纵cüoz?ng 416 草cǎo 420 册 ca 424 测ca428 策略 cal?a 432 曾经 c?ngj?ng 436 差价chüjià 440 茶chá 444 察chá448 差点儿chàdiǎnr 452 缠chán456 产品chǎnpǐn 460 产值chǎnzhí 464 长cháng468 长短chángduǎn 472 长征chángzhyng476 尝试chángshì 480 常识chángshí 484 场chǎng488 场所chǎngsuǒ 492 超出chüochū 496 巢cháo 500 潮流cháoliú 504 车 chy 508 车站chyzhàn512 撤cha513 撤销 chaxiüo 517 沉淀ch?ndiàn 521 沉重 ch?nzh?ng 525 陈述ch?nshù 529 称号chynghào 533 成ch?ng537 成功ch?nggōng 514 臣 ch?n 515 尘 ch?n 516 沉 ch?n518 沉积 ch?nj? 522 沉着 ch?nzhu? 526 称chan530 称呼 chynghu 534 成本 ch?ngbtn 538 成果ch?ngguǒ 519 沉默 ch?nm? 523 陈 527 趁ch?n chan520 沉思 ch?ns? 524 陈旧ch?njiù 528 称 532 撑chyng chyng531 称赞chyngzàn 535 成虫 ch?ngch?ng 539 成绩ch?ngjì 536 成分 ch?ng?fan 540 成就ch?ngjiù 541 成立 545 成为 549 成长 553 诚恳 557 承担 561 城市 565 乘客 569 程式 573 吃 577 池581 持久 585 齿589 翅膀 593 冲破 597 充分 601 虫 605 重新 609 抽 613 丑617 出产 621 出口 625 出门 629 出生 633 出现 637 初级ch?nglì ch?ngw?i ch?ngzhǎng ch?ngktn ch?ngdün ch?ngshì ch?ngka ch?ngshì ch? chíchíjiǔ chǐchìbǎng chōngp? chōngfan ch?ng ch?ngx?nchōu chǒuchūchǎn chūkǒu chūm?n chūshyng chūxiàn chūjí542 成年ch?ngnián 546 成效ch?ngxiào 550 呈ch?ng554 诚实ch?ng?shí 558 承认 ch?ngran 562 城镇 ch?ngzhan 566 盛ch?ng570 程序ch?ngxù 574 吃饭ch?fàn 578 池塘chítáng 582 持续chíxù 586 赤chì 590 冲chōng594 冲突chōngtū 598 充满chōngmǎn 602 重zh?ng606 崇拜ch?ngbài 610 抽象chōuxiàng 614 臭ch?u618 出发chūfü 622 出来chū?lái 626 出去chū?qù 630 出售chūsh?u 634 出血chūxit 638 初期chūq?543 成人 ch?ngr?n 547 成语ch?ngyǔ 551 呈现ch?ngxiàn 555 承ch?ng559 承受 ch?ngsh?u 563 乘 ch?ng 567 程ch?ng571 惩罚ch?ngfá 575 吃惊 ch?j?ng 579 迟chí 583 尺chǐ587 赤道chìdào 591 冲动chōngd?ng 595 充chōng599 充实chōngshí 603 重复ch?ngfù 607 崇高ch?nggüo 611 仇恨 ch?uhan 615 出chū 619出发点chūfüdiǎn 623 出路chūlù 627 出色chūsa 631 出土chūtǔ 635 初chū639 初中chūzhōng544 成熟 548 成员 552 诚 556 承包 560 城564 乘机 568 程度 572 秤 576 吃力 580 持584 尺度 588 翅592 冲击 596 充当 600 充足 604 重合 608 冲 612 愁616 出版 620 出国 624 出卖 628 出身 632 出席 636 初步 640 除ch?ngshú ch?ngyuán ch?ng ch?ngbüoch?ngch?ngj? ch?ngdù chang ch?lì chíchǐdù chìchōngj? chōngdüng chōngzú zh?ngh? ch?ng c h?uchūbǎn chūgu? chūmài chūshyn chūxí chūbù chú感谢您的阅读，祝您生活愉快。

汉字五笔编码速查表.pdf综述汉字五笔编码速查总表使用说明：1．本汉字五笔字型编码速查总表共有 6763 个，总表按照汉字汉语拼音排序。

2．每个汉字的右方是五笔字型的键位编码。

编码右侧标注②的，表示该汉字为二级简码汉字；编码右侧标注③的，表示该汉字为三级简码汉字。

3．本表包括五笔字型 98 版汉字的键位编码，当 98 版编码与 86 版编码不同时，本表在86版编码下方单独列出98版汉字编码，并在98版编码右上角标注“98”。

a吖KUHH③口丨丨KUHH98口丨丨阿BSKG②阝丁口一啊KBSK②口阝丁口锕QBSK③钅阝丁口嗄KDHT口ǐ目夂ai哎KAQY③口艹KARY98③口艹哀YEUú冫唉KCTD③口厶大埃FCTD③土厶大挨RCTD③扌厶大锿QYEY钅ú 捱RDFF扌厂土土皑RMNN白山己乙RMNN98③白山己乙癌UKKM③疒口口山嗳KEPC③口ǒ冖又矮TDTV大禾女蔼AYJN③艹讠日乙霭FYJN③雨讠日乙艾AQU艹冫ARU98艹冫爱EPDC②ǒ冖ì又EPDC98ǒ冖ì又砹DAQY石艹DARY98石艹隘BUWL③阝八皿嗌KUWL③口八皿嫒VEPC女ǒ冖又VEPC98③女ǒ冖又碍DJGF③石日一寸暧JEPC③日ǒ冖又瑷GEPC王ǒ冖又an安PVF②宀女二桉SPVG③木宀女一氨RNPV③乙宀女RPVD98 气宀女三庵YDJN 广大日乙ODJN98③广大日乙谙YUJG③讠立日一鹌DJNG 大日乙一鞍AFPV③廿ǖ宀女俺WDJN 亻大日乙埯FDJN③土大日乙铵QPVG③钅宀女一揞RUJG 扌立日一犴QTFH 丿干丨岸MDFJ 山厂干按RPVG③扌宀女一案PVS③宀女木胺EPVG③月宀女一暗JUJG②日立日一黯LFOJ 土灬日ang肮EYMN③月亠几乙EYWN98③月亠几乙昂JQBJ③日卩盎MDLF③冂大皿二ao凹MMGD 几冂一三HNHG98 丨乙丨一坳FXLN③土幺力乙坳FXET98③土幺力丿敖GQTYǘ勹夂嗷KGQT口ǘ勹夂廒YGQT广ǘ勹夂OGQT98③广ǘ勹夂獒GQTDǘ勹夂犬遨GQTPǘ勹夂辶熬GQTOǘ勹夂灬翱RDFN白大十羽聱GQTBǘ勹夂耳螯GQTJǘ勹夂虫鳌GQTGǘ勹夂一鏖YNJQ广金OXXQ98 匕匕金袄PUTD③冫丿大媪VJLG③女日皿一岙TDMJ③丿大山傲WGQT亻ǘ勹夂奥TMOD③丿冂米大骜GQTCǘ勹夂马GQTG98ǘ勹夂一澳ITMD③氵丿冂大懊NTMD③忄丿冂大鏊GQTQǘ勹夂金赘GQTMǘ勹夂贝ba八WTY八丿丶巴CNHN③巴乙丨乙叭KWY口八丶吧KCN②口巴乙岜MCB山巴ē芭ACB②艹巴ē疤UCV疒巴巛捌RKLJ扌口力刂RKEJ98扌口力刂2巧记字根学五笔笆TCB 巴ē粑OCN 米巴乙拔RDCY③扌ō又丶RDCY98③扌又丶茇ADCU③艹ō又冫ADCY98③艹又丶菝ARDC③艹扌ō又ARDY98③艹扌丶跋KHDC 口止ō又KHDY98 口止丶魃RQCC 白儿厶又RQCY98 白儿厶丶把RCN 扌巴乙钯QCN 钅巴乙靶AFCN③廿ǖ巴乙坝FMY 土贝丶爸WQCB③八巴ēWRCB98③八巴ē罢LFCU③土厶冫鲅QGDC 一ō又QGDY98 一丶霸FAFE③雨廿ǖ月灞IFAE③氵雨廿月耙DICN③三小巴乙FSCN98③二木巴乙bai掰RWVR 手八刀手白RRRR③（键名汉字）佰WDJG③亻ǐ日一柏SRG 木白一捭RRTF③扌白丿十摆RLFC③扌土厶败MTY 贝夂丶MTY98②贝夂丶拜RDFH ī三十丨稗TRTF 禾白丿十TRTF98③禾白丿十ban扳RRCY③扌又丶班GYTG③王丿王般TEMC③丿ó几又TUWC98 丿ó几又颁WVDM③八刀ǐ贝斑GYGG③王文王一搬RTEC③扌丿ó又RTUC98③扌丿ó又瘢UTEC 疒丿ó又UTUC98 疒丿ó又癍UGYG③疒王文王UGYG98 疒王文王阪BRCY 阝又丶坂FR CY③土又丶板SRCY③木又版THGC 丿丨一又钣QRCY③钅又丶2 舨TERC 丿ó又TURC98 丿ó又办LWI②力八氵EW98 力八半UFK②十UGK98②伴WUFH③亻十丨WUGH98 丨扮RWVN③RWVT98 扌八刀丿拌RUFH 扌十丨RUGH98 丨绊XUFH③纟十丨XUGH98③丨瓣URCU③bang邦DTBH③三丿阝丨帮DTBH②三丿阝丨DTBH98 三丿阝丨梆SDTB③木三丿阝浜IRGW 氵斤一八IRWY98③氵丘八丶绑XDTB③纟三丿阝榜SUPY③木冖方SYUY98③木亠方膀EUPY③月六冖方EYUY98③月亠方傍WUPY③亻冖方WYUY98 亻亠方谤YUPY③讠冖方YYUY98③讠亠方棒SDWH③木三人丨SDWG98 木三八蒡AUPY 艹冖方AYUY98 艹亠方磅DUPY③石冖方DYUY98③石亠方镑QUPY③钅冖方QYUY98③钅亠方bao包QNV②勹巳巛孢BQNN③子勹巳乙苞AQNB③艹勹巳ē胞EQNN③月勹巳乙煲WKSO 亻口木火龅HWBN 止人凵巳褒YWKE③亠亻口ú雹FQNB③雨勹巳ē宝PGYU③宀王丶冫饱QNQN 乙勹巳保WKSY②亻口木鸨XFQG③匕十勹一堡WKSF 亻口木土葆AWKS③艹亻口木褓PUWS 冫亻木报RBCY②扌卩又抱RQNN③扌勹巳乙豹EEQYǒù勹丶EQYY98③豸勹丶丶趵KHQY口止勹丶鲍QGQN③一勹巳暴JAWI③日ā八水爆OJAI③火日ā水bei呗KMY口贝丶陂BHCY③阝又丶BBY98阝皮丶卑RTFJ白丿十RTFJ98③白丿十杯SGIY③木一小丶SDHY98③木ǐ卜丶悲DJDN三三心HDHN98③丨三丨心碑DRTF③石白丿十鹎RTFG白丿十一北UXN②丬匕乙贝MHNY贝丨乙丶狈QTMY丿贝丶QTMY98③丿贝丶邶UXBH③丬匕阝丨备TLF夂田二TLF98②夂田二背UXEF③丬匕月二钡QMY钅贝丶倍WUKG③亻立口一悖NFPB忄十冖子被PUHC冫又PUBY98③冫皮丶惫TLNU③夂田心冫焙OUKG③火立口一OUKG98火立口一辈DJDL三三车HDHL98丨三丨车碚DUKG③石立口一蓓AWUK艹亻立口褙PUUE冫丬月鞴AFAE廿ǖā用鐾NKUQ尸口辛金庳YRTF③广白丿十ORTF98③广白丿十孛FPB十冖子铸QDTF③钅三丿寸ben奔DFAJ③大十廾贲FAMU③十艹贝冫锛QDFA③钅大十廾本SGD②木一三苯ASGF③艹木一二畚CDLF③厶大田二坌WVFF 八刀土二WVFF98③八刀土二笨TSGF③木一二beng蚌JDHH③虫三丨丨崩MEEF③山月月二绷XEEG③纟月月一嘣KMEE③口山月月KMEE98 口山月月甭GIEJ③一小用DHEJ98 卜用刂泵DIU 石水冫迸UAPK③廾辶甏FKUN 土口乙FKUY98 士口丶蹦KHME 口止山月KHME98③口止山月bi逼GKLP 一口田辶荸AFPB 艹十冖子鼻THLJ③丿目田匕XTN 匕丿乙比XXN②匕乙吡KXXN③口匕乙KXXN98 口匕乙妣VXXN③女匕乙彼THCY③彳又TBY98 彳皮丶秕TXXN③禾匕乙俾WRTF③亻白丿十笔TTFN②丿二乙TEB98 毛ē舭TEXX③丿óTUXX98 丿ó鄙KFLB③口十口阝币TMHK③丿冂丨必NTE②心丿彡毕XXFJ③匕十闭UFTE③门十丿彡庇YXXV③广匕巛OXXV98 广匕匕巛畀LGJJ③田一刂哔KXXF 口匕十毖XXNT 匕心丿荜AXXF 艹匕十陛BXXF②阝匕土毙XXGX 匕一匕狴QTXF 犭丿土铋QNTT 钅心丿丶婢VRTF③女白丿十敝UMIT③冂小夂ITY98 攵丶萆ARTF③艹白丿十弼XDJX 弓ǐ日弓愎NTJT 忄日夂筚TXXF 匕十滗ITTN③氵丿乙ITEN98 氵毛乙痹ULGJ 疒田一蓖ATLX③艹丿口匕裨PURF③冫白十跸KHXF 口止十辟NKUH③尸口辛丨弊UMIA 冂小廾ITAJ98 攵廾刂碧GRDF③王白石二箅TLGJ③田一蔽AUMT③艹冂夂AITU98 艹攵冫壁NKUF 尸口辛土嬖NKUV 尸口辛女篦TTLX 丿口匕薜ANKU③艹尸口辛避NKUP②尸口辛辶濞ITHJ 氵丿目臂NKUE 尸口辛月髀MERF 月白十璧NKUY 尸口辛丶襞NKUE 尸口辛úbian边LPV②力辶巛EP98 力辶砭DTPY③石丿之笾TLPU③力辶冫TEPU98③力辶冫编XYNA 纟丶尸艹煸OYNA 火丶尸艹蝙JYNA 虫丶尸艹鳊QGYA 一丶艹鞭AFWQ③廿ǖ亻AFWR98 廿ǖ亻贬MTPY③贝丿之扁YNMA 丶尸冂艹窆PWTP 宀八丿之匾AYNA 匚丶尸艹碥DYNA 石丶尸艹褊PUYA 冫丶艹卞YHU 亠卜冫弁CAJ 厶艹忭NYHY 忄亠卜丶汴IYHY③氵亠卜丶苄AYHU③艹亠卜冫便WGJQ③亻一日WGJR98 亻一日变YOCU②亠又冫YOCU98③亠又冫缏XWGQ 纟亻一XWGR98 纟亻一遍YNMP③丶尸冂辶辨UYTU③辛丶丿辛UYTU98 辛丶丿辛汉字五笔编码速查表 3辩UYUH③辛讠辛丨辫UXUH③辛纟辛丨biao彪HAME ê七几彡HWEE98③虍几彡彡标SFIY③木二小丶飑MQQN 几勹巳WRQN98 几勹巳髟DET í彡丿骠CSFI③马西二小CGSI98③一西小膘ESFI③月西二小瘭USFI③疒西二小镖QSFI③钅西二小飙DDDQ 犬犬犬DDDR98 犬犬犬飚MQOO③几火火WROO98 几火火镳QYNO 钅广灬QOXO98③钅匕灬表GEU②ǘú冫婊VGEY 女ǘú丶裱PUGE 冫ǘú鳔QGSI③一西小bie憋UMIN 冂小心ITNU98 攵心冫鳖UMIG 冂小一ITQG98 攵一别KLJH③口力刂丨KEJH98③口力刂丨蹩UMIH ɑITKH98 攵口止瘪UTHXbin宾PRGW③宀斤一八PRWU98③宀丘八冫彬SSET③木木彡丿傧WPRW③亻宀八斌YGAH③文一弋止滨IPRW③氵宀八缤XPRW③纟宀八槟SPRW③木宀八镔QPRW③钅宀八濒IHIM 氵止小贝IHHM98 氵止贝豳EEMK 豕豕山MGEE98③山一摈RPRW③扌宀八殡GQPW③一夕宀八GQPW98 一夕宀八膑EPRW③月宀八髌MEPW 月宀八鬓DEPW í彡宀八。

五笔字型字根表
五笔字型字根表11 G 王旁青头兼五一
12 F 土士二干十寸雨
13 D 大犬三羊古石厂
14 S 木丁西
15 A 工戈草头右框（匚）七
21 H 目具上止卜虎皮
22 J 日早两竖与虫依
23 K 口与川，字根稀
24 L 田甲方框四车力
25 M 山由贝，下框骨头几
31 T 禾竹一撇双人立，反文条头共三一
32 R 白手看头三二斤
33 E 月彡（衫）乃用家衣底
34 W 人和八，三四里
35 Q 金勺缺点无尾鱼，犬旁留叉儿一点夕，氏无七
41 Y 言文方广在四一，高头一捺谁人去
42 U 立辛两点六门病
43 I 水旁兴头小倒立
44 O 火业头，四点米
45 P 之宝盖，摘示衣
51 N 已半巳满不出己，左框折尸心和羽
52 B 子耳了也框向上
53 V 女刀九臼山朝西
54 C 又巴马，丢矢矣
55 X 慈母（纟）无心弓和匕，幼无力。