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International Journal of Control
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Constrained agreement protocols for tree graph
topologies
Rahul Chipalkatty a & Magnus Egerstedt a
a Georgia Institute of T echnology, Atlanta, GA 30332, USA
Available online: 07 Feb 2012
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International Journal of Control 2012,1–18,iFirst
Constrained agreement protocols for tree graph topologies
Rahul Chipalkatty *and Magnus Egerstedt
Georgia Institute of Technology,Atlanta,GA 30332,USA (Received 11March 2011;final version received 7January 2012)
This article presents a novel way of manipulating the initial conditions in the consensus equation such that a constrained agreement problem is solved across a distributed network of agents,particularly for a network represented by a tree graph.The presented method is applied to the problem of coordinating multiple pendula attached to mobile bases.The pendula should move in such a way that their motion is synchronised,which calls for a constrained optimal control problem for each pendulum as well as the constrained agreement problem across the network.Simulation results are presented that support the viability of the proposed approach as well as a hardware demonstration.
Keywords:consensus algorithm;optimal control;graph-based multi-agent control;constrained agreement protocol
1.Introduction
The agreement protocol (or consensus equation)has by now emerged as a standard way to achieve agreement among agents in a distributed network.It can be utilised for anything from agreement in embedded physical systems like mobile robots or UAVs,to distributed computer networks, e.g.Tanner,Jadbabaie,and Pappas (2000),Ren and Beard (2004),Dimarogonas and Kyriakopoulos (2007),Olfati-Saber and Murray (2003),Jadbabaie,Lin,and Morse (2003)and Mesbahi and Egerstedt (2010).And,as the value on which the agents agree is dependent on the agents’initial conditions,it is not overly surprising that this dependency can be employed such that the final agreement state is guaranteed to satisfy certain constraints.
This work seeks to address the problem of finding an agreement state that satisfies a global state con-straint given only local interactions among distributed agents.Specifically,within a group of agents,con-straints exist between certain pairs of agents together with an information exchange,resulting in what is referred to in this article as pairwise constraints.The pairwise constraints naturally lead to a network topology where nodes represent agents and edges represent pairwise constraints and information exchange.
The pairwise constraints that involve a particular agent are called that agent’s local constraints and this particular agent only exchanges information with the agents with which it shares a pairwise constraint,hence the ‘local’descriptor.Subsequently,the global con-straint for the entire group is the constraint that all agents satisfy their respective local constraints.Each agent can communicate its own state as well as its ‘opinion’of what the other agents’states should be (i.e.state opinions)such that the entire network’s pairwise constraints are met.
As such,the goal of this work is to utilise the consensus equation to arrive at an agreement on the state opinions of every agent in the network such that every pairwise constraint in the network is satisfied (global constraint)given that only the local constraints are initially satisfied.
We apply this constrained agreement protocol to the problem of controlling a collection of cart-driven pendula in a coordinated fashion.In particular,their mobile bases are to be controlled in such a way that,at some specified terminal time,they move in unison (with the same frequency and phase)at a fixed inter-pendula distance.This should be achieved using only local information,i.e.the control actions are only to be controlled based on information from neighbouring pendula.
The problem is ensuring that the agreement proto-col results in final state opinions for all pendula such that these final states satisfy the global constraint.Although running the standard consensus equation (e.g.Jadbabaie et al.2003;Olfati-Saber and Murray 2003)–or versions of the gossip algorithm Boyd,
*Corresponding author.Email:chipalkatty@https://www.doczj.com/doc/7a5207829.html,
ISSN 0020–7179print/ISSN 1366–5820online ?2012Taylor &Francis
https://www.doczj.com/doc/7a5207829.html,/10.1080/00207179.2012.656288https://www.doczj.com/doc/7a5207829.html,
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Ghosh,Prabhakar,and Shah (2005)and Xiao,Boyd,
and Lall (2005)–will result in an agreement,the agreed upon states are not guaranteed to satisfy the global constraint.However,in this work,we will demonstrate that the initial conditions for the agreement protocol can be manipulated such that the resulting state opinions will satisfy the global constraints.Relevant work on agreement protocols for systems with oscil-lating dynamics include (Jadbabaie,Motee,and Barahona 2004).The authors investigate the stability of the Kuromoto model for coupled nonlinear oscilla-tors;however,they do not address constraints on the agreement state.
Although optimal control techniques are used in the control formulation presented in this article,the problem is,at heart,a distributed agreement problem rather than a distributed optimal control problem.For more on the latter problem see,for example,Rotkowitz and Lall (2006),Motee and Jadbabaie (2008),Bamieh,Paganini,and Dahleh (2002),Rantzer (2009)and the references therein.These works seek to solve distributive optimal control problems over a network while the work presented in this article simply utilises optimal control to drive the system to a desired state as determined through an agreement protocol.Any form of control could,in principle,have been used here.However,optimal control provides an effective method for solving the types of constrained problems under consideration in this article.
Relevant works on constrained agreement proto-cols include Moore and Lucarelli (2005)and Nedic,Ozdaglar,and Parrilo (2008).Moore and Lucarelli (2005)address the problem of leader-driven agreement where the agreement state can be driven to a set-point and thus agreement states can be driven to satisfy hard constraints.In Nedic et al.(2008),the authors present agreement protocols where the agreed upon states are constrained to lie in a convex set.In both of these works,the resulting algorithms require that the con-sensus update law be modified over time with time-varying weights.In contrast to this,Chipalkatty,Egerstedt,and Azuma (2009)presents a simple,static update law for achieving agreement while satisfying the constraints for a line topology network.In this article,we will present a novel method of using a static update law to satisfy global constraints over not just a line topology,but any arbitrary directed tree graph topol-ogy.The novelty of this method is that it only requires a priori knowledge of the graph topology and local information exchange to initialise the agreement state.This results in a static update law,using local infor-mation,to reach an agreement state that satisfies a global constraint.We will also present the viability of this approach through both simulation and hardware demonstration.
The outline of this article is as follows:in Section 2,we introduce the distributed network of agents and its constraints,followed by a discussion in Section 3about the constrained consensus problem.In Section 4,we show how selecting the initial conditions appropriately leads to an agreement state that satisfies the global constraints and in Sections 5and 6,we show how this method is applied to the networked pendulum exam-ple.Simulation results are presented in both Sections 5and 6as well as hardware results in Section 5.2.Problem formulation 2.1Notation
A graph,G ,is defined by a node set,V ?{1,2,3,...,N }of N nodes and an edge set E &V ?V of unordered node pairs.Two nodes,i and j ,are adjacent,or neighbours,if (i ,j )2E .The neighbourhood set of a node i 2V ,N i ,is the set of nodes j 2V adjacent to node i .A tree graph is a specific type of graph in which there exists only one path from a particular node to any other node in the graph.Note that for tree graphs,the number of edges,M ,in the graph is given by M ?N à1.Edges can be given an orientation using :E !{à1,1},resulting in a directed graph,G ,for which an associated incidence matrix,D ?[D ij ]2R N ?M ,has elements given by
D ij ?1
if vertex i is the tail of edge j ,à1if vertex i is the head of edge j ,
0otherwise,8
><>:e1Tfor i ?1,...,N and j ?1,...,M .2.2Multi-agent network
The multi-agent network in this article is modelled by a
tree graph,G ?(V ,E )where the N nodes correspond to N agents.The M edges of the network correspond to information exchange between agents as well as a pairwise constraint.Each edge is assigned an arbitrary orientation,as given by ,resulting in an associated incidence matrix,D .Note,however,that the graph is still an undirected one,so the information exchange is undirected.Representing the network through an incidence matrix is key here because the incidence matrix will be utilised to construct formal definitions of the local and global constraints.
In order to illustrate the operations used later,an example network is presented and the prescribed operations will be performed on this network through-out this article.2.2.1Example graph
The example tree graph,^G
?e^V ,^E Twith ^V ?f 1,2,3,4,5g and ^E
?{(1,2),(1,4),(4,3),(4,5)},is given 2R.Chipalkatty and M.Egerstedt
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with arbitrary edge orientations as presented in
Figure 1.The corresponding incidence matrix,^D
,is ^D ?1100
à10
0000à100à11
1000à1
2
6
66
66
6
437777775:e2T2.3States and opinions
Each agent in the network has an associated state and we let x ii 2R l denote agent i ’s state for all i 2V .The double indices are used here to denote agent i ’s opinion of itself,which coincides with its state,and this notation is now used again to denote an agent’s opinions of other agents.
Each agent in the network will form a state opinion of all the other agents in the network and share those opinions with its neighbours.Let x ij (t )2R l be agent i ’s opinion of what agent j ’s state should be such that all the pairwise constraints in the network are satisfied (i.e.agent i ’s’state opinion of agent j ).The collection of these opinions is denoted by the state opinion vector,x i ,for each agent,where x i ?[x i 1,...,x ii ,...x iN ]for all i 2V .A form of this vector will eventually become the quantity used in the proposed agreement protocol.
2.4Pairwise constraints
Let each edge with arbitrary orientation denote an associated linear pairwise constraint on agents i and k for all nodes i ,k 2V where (i ,k )2E .This linear pairwise constraint between agents is defined as
P ex ii àx kk T?b ,
e3T
with vertex i being the tail of the edge,b 2R p and P 2R p ?l .Note that P has full row rank.This particular form for the pairwise constraints was chosen because it is linear with respect to the difference in the agent’s states,allowing for relative constraints, e.g.for synchronising movement between the agents.
2.5Constraint definition
Given a pairwise constraint for every edge,the global
constraint for the network is defined as the collection of all pairwise constraints and is satisfied when the state of every node in the network satisfies all pairwise constraints.An agent’s local constraints are defined as the set of all pairwise constraints it shares with its neighbours.When a particular node’s state satisfies all the pairwise constraints with its neighbours,this agent’s state satisfies its local constraints.
The following is a discussion on how to construct the local constraints for each node and the global constraints for the network given the graph,G ,and the incidence matrix,D .Since the incidence matrix gives us a representation of all the edges in the network,the incidence matrix will first be utilised to construct an expression for the global constraint on the network.
Let x g 2R lN be a state opinion vector such that all its state opinions satisfy all the pairwise constraints in the network.Then,the global constraint is defined as
eD T P Tx g ?1 b ,
e4T
where denotes the Kronecker product and 1is an M -dimensional vector with all entries being 1.
From the example network G ,the global con-straint,(D T P )x g ?1 b ,is satisfied if,for x g ?[x 11,...,x 55]T ,
P àP 000P 00àP 000àP P 0000P àP
2666437775x 11
x 22x 33x 44x 55266666643
7
777775
?b b b b 2666437775:e5THere,since each row corresponds to an edge,this expression represents every pairwise constraint in the network.The following operations isolate the rows of the transposed incidence matrix that correspond to edges that node i is incident with and apply the pairwise constraints to these edges to arrive at an expression for the local constraints of agent i .
For all i 2V ,we define the diagonal matrix,F i
2R M ?M ,whose m th diagonal element is 1if D im ?0and 0otherwise,as well as the complement,F ic 2R M ?M ,which is a diagonal matrix whose m th diagonal element is 0if D im ?0and 1otherwise,for i ?1,...,N and m ?1,...,M .To illustrate this fur-ther,the F i and F ic matrices are shown for agent 4for
the example graph ^G
as ^F 4?00000100001000
01
2666437775,^F 4c ?10000000000000
00
2666437775
:Figure 1.Tree graph example,^G
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Note ^F
4contains a 1in the column or row corre-sponding to the edges agent 4is incident with and vice
versa for ^F
4c Then,let T i 2R M ?N be defined as
T i ?F i D
T
with the compliment,
T c i ,
being
T c i ?F ic D T :
Note that F i tF ic ?I ,where I is the M ?M identity matrix.Again,the example is worked further to show the matrices resulting from these operations.T i and T c i for agent 4are
^T 4?00000100à1000à1100001à1
2666437
775
,
^T c 4?1à1000000000
00000
00
266643
77
75:
Lemma 2.1:Given matrices T i and T c i for every agent i in the network for i ?1,...,N ,then T c i tT i ?D T .Proof:
T c i tT i ?F i D T tF ic D T
?eF i tF ic TD T ?ID T ?D T :
?
Here,the rows of T i 2R M ?N encode only the edges that node i is incident with while the remaining rows have all elements as zero.Similarly,the rows of T c i encode the edges that vertex i is not incident with and edges incident with node i have all zero elements.
Additionally,a vector is needed to equate the appropriate constraints in each row to b .In other words,each non-zero row in (T i P )x must equal b .
The vector,f i ??f i m M
m ?12R M ,is defined as having its m th element be 1if T im ?0or 0otherwise,for i ?1,...,N and m ?1,...,M .Continuing with our example network,^f 4??0111 T .
Lemma 2.2:Given the vector ,f i ,for every agent i in the network for i ?1,...,N ,then
X N i ?1
f i ?21:
Proof:For every edge j in the network,there are two agents incident with that edge.Therefore,a 1will appear in the j -th element for two agents,resulting in a
2in every element of the sum of f i ’s.
?The matrix,T i ,can now be used to define the local constraints on agent i .Let x i satisfy pairwise con-straints that involve agent i ,such that
eT i P Tx i ?f i b ,
e6T
for i ?1,...,N .In other words,only the pairwise constraints agent i can satisfy are represented in (6).The local constraints for agent 4in the graph example ^G
,would be satisfied if,for x 4?[x 41,...,x 45]T ,e^T
4 P Tx 4?^f 4 b or 00000P 00àP 000àP P 0000P àP
2666437775x 41
x 42x 43x 44x 452
6666
66437777775
?0b b b 2666437775
:Note that in the example,agents 2and 4are not neighbours so agent 4has no basis on which value to assign to x 42.Also note that this value is not required for agent 4’s local constraints to be satisfied.However,in order to utilise the consensus equation,these agents must assign state opinions to non-adjacent agents.
In summary,each agent assigns a state opinion value to neighbours based on the initial state values communicated such that they meet the local con-straints,(6).Then,the consensus equation will be used to reach an agreement on what the state opinion vectors should be.This agreed upon state opinion vector needs to satisfy the global constraint,(4).So the problem,here,is to determine how each agent should initialise their state opinion vectors so that the global constraint is satisfied by the result of the consensus equation.
3.Applying the consensus equation
Again,the goal of this work is to find a state opinion
vector that satisfies the global constraint among distributed agents using only local interactions.In order to accomplish this,the consensus algorithm will be used to have the agents come to an agreement on such a state.It should be noted that each agent’s initial state opinions will satisfy the local constraints for that agent only.
Recall x i is the quantity over which the consensus equation will run and it is desired that the resulting quantity satisfies the global constraint.For agents in the neighbourhood of agent i ,the initial state opinions are made based on the states of the agent as commu-nicated by those agents.However,for agents not in the neighbourhood of agent i ,state opinions are still required to construct the state opinions vector and utilise the consensus equation.However,the lack of
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communication means that initialising the state opin-ion vector with state opinions for non-adjacent agents is a problem.The following section discusses how to deal with this problem by first simply setting these values to zero and showing why the resulting agree-ment state does not satisfy the global constraint.
3.1Attempt 1:filling the state opinion vector with 0’s
For undirected connected graphs,the consensus equa-tion,as given by
_x i et T?àX
k "N i
ex i et Tàx k et TTfor i ?1,...,N ,e7T
converges to the invariant centroid,x c 2R Nl ,where
x c ?1N X N
i ?1
x i e0T
e8T
(see,e.g.Mesbahi and Egerstedt 2010).Note that the consensus algorithm requires that agents communicate their state opinion vectors with their neighbours and vice versa.
Suppose the initial state opinions of non-adjacent agents are simply set to https://www.doczj.com/doc/7a5207829.html,ing the example graph,it will be shown that the resulting agreement state does not satisfy the global constraint although the initial conditions of each agent’s state opinion vector satisfy that agent’s local constraints.Let the initial state opinion vectors be
x 1?x 11x 120x 1402666666437777775,x 2?x 21x 220002666666437777775,x 3?0
0x 33x 340
266666643
777
77
75,x 4?x 410x 43x 44x 452666666437777775,x 5?000x 54x 55266666643777
7775:So,the resulting agreement state is
x c ?x 11
tx 21
tx
41
5
x 12tx 225
x 33tx 435
x 14tx 34tx 44tx 545
x 45tx 55
5
2666
6
666
66666
643
77777777777
775:
e9T
Given that the initial state opinion vectors satisfy the local constraints,
e^T 1 P Tx 1?^f 1 b ?Px 11àPx 12
Px 11àPx 140
2666437775?b b 00
266643
7
775,
e10T
e^T 2 P Tx 2?^f 2 b ?Px 21àPx 22
00
2
666437775?b 000
266643
7775,e11T
e^T 3 P Tx 3?^f 3 b ?00Px 34àPx 330
2
66643
7775?0
0b 0
2666437775
,e12T
e^T 4 P Tx 4?^f 4 b ?0
Px 41àPx 44Px 44àPx 43Px 44àPx 45
2
666437775?0b b b
266643
7
775
,e13T
and
e^T 5 P Tx 5?^f 5 b ?000
Px 54àPx 55
2
6664
3
7775?000b 2666437775:e14T
The result of plugging values from the local constraints into the global constraint is that
e^D T P Tx c ?25b t1
Px 41
b t12P ex 21àx 34àx 54Tb t12P ex 14tx 54Tb t1
2
P ex 14tx 34T
2
66666666664377777777
7
75?b b b b 2666437775e15Tand therefore the agreement state does not satisfy the global constraint.
As a result,it is proposed that local information (i.e.the state opinion vector shared by an agent’s
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neighbours)be used to initialise the state opinions for
non-adjacent agents such that the extra terms in each row of (15)will be cancelled,i.e.x 41in row 1,x 21,x 34,x 54in row 2,x 14,x 54in row 3and x 14,x 34in row 4.In addition,a gain value will be needed to cancel the 25term in (15).By doing this,it will be shown that the resulting agreement state satisfies the global constraint.
3.2Attempt 2:initial condition definition by
propagation
It is proposed that the state opinions of agents not adjacent to agent i should be assigned values in x i such that the equation
eT c i P Tx i ?0M
e16T
is satisfied,where 0M is an M -dimensional vector with all elements being zero.Note that this requires that each agent a priori knows the structure of the network in the form of the incidence matrix.
For each agent,this equation is sufficient to choose state opinion values for every non-adjacent agent.For any agent i ,there are d i agents adjacent to agent i .Therefore,there are N à1àd i agents not adjacent to agent i,which implies (N à1àd i )l state opinions are undefined.T c i contains (M àd i )l rows that represent pairwise constraints.This results in (M àd i )l equations and (N à1àd i )l unknowns.
As a result,effectively choosing initial state opin-ions is feasible only if M ?N à1,i.e.G is a tree graph.For graphs with cycles,M 4N à1,therefore,the system of equations is overdetermined.For incomplete graphs,M 5N à1,resulting in a system of equations that is insufficient to solve for all the initial state opinions.
To give further insight on the resulting state opinion assignments,we will return to the example
graph,G .The intermediate matrix T c
4results in
T c 4?1à100000000000000000026643775:So,we assign non-adjacent agents state opinions according to
eT c
4 P Tx 4?P ex 41àx 42T
00
2666437775?0000
266643
7775:
x 41is a communicated value because agent 1is
adjacent to agent 4,so x 42is initially assigned the same value as x 41,i.e.it is assigned the value of the
agent adjacent to agent 4that is in the path from agent 2to agent 4.Similarly for agent 5,
eT c
5 P Tx 5?P ex 51àx 52T
P ex 51àx 54TP ex 54àx 53T0
2666437775?0000
266643
7
77
5
revealing that,as x 54is defined through communica-tion,x 51and x 53are set to x 54.As a result,x 52is also set to the value of x 54.Figures 2and 3show this graphically for agents 4and 5.Nodes of the same hatching denote that the initial state opinions are equal from the view of the solid coloured node.
In other words,the state opinion assignments propagate out into the network from the neighbour-hood of agent i as shown in Figure 4.By setting the state opinion of a non-adjacent agent j to the same value an adjacent agent on the path from agent i to the non-adjacent agent j ,we are guaranteeing that the pairwise constraints on that agent results in 0l in agent i ’s opinion.
As a result,each element of x i is defined and it will now be shown that this choice of initial conditions for the consensus equation will result in an agreement state that satisfies the global constraint.The quantity used in the consensus equation is now denoted,X i (0)? x i for i ?1,...,N with X i 2R Nl and gain 2R .The consensus equation,
_X
i et T?àX
k "N i
eX i et TàX k et TTfor i ?1,...N e17
TFigure 3.Initial conditions propagation for example graph agent 5.The state opinion for non-adjacent agents 1,2and 3are assigned the value of adjacent agent
4.
Figure 2.Initial conditions propagation for example graph agent 4.The state opinion for non-adjacent agent 2is assigned the same value as adjacent agent 1.
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results in
X c ?1N X N i ?1X i e0T? N X N i ?1
x i
e18T
as the final agreement state,X c 2R Nl .
It is required that this agreement state satisfies the global constraint where each agent’s state opinion vector satisfies the pairwise constraint for each edge in the graph.Therefore,it is needed that
eD T P TX c ?1 b :
e19T
To verify that the agreement state satisfies the global constraint,we can plug in the RHS of (18)for X c and rewrite the global constraint,(19),as
eD T
P TX c ?eD T
P T N X
N i ?1
x i
? N X N i ?1
eD T P Tx i :e20T
Using (6),(20)can be written as
? N X N i ?1
eeT i tT c i T P Tx i :e21T
By the distributive property of the Kronecker product and (6),(21)becomes
?
N X N i ?1eT i P Tx i t N X
N i ?1eT c i P Tx i ? N X N i ?1f i b t N X N i ?1
eT c i P Tx i :e22T
Recalling Lemma 2.2,(22)is rewritten as
? N 21 b t N X N
i ?1
eT c i P Tx i :e23T
Plugging (16)into (23)results in
eD T P TX c ?
21 b :Plugging in ?N 2,it is shown that
eD T P TX c ?1 b :
Therefore,the global constraint is satisfied by the final agreement state through the manipulation of the initial conditions of the consensus equation.This leads us to the following theorem.
Theorem 3.1:If the state opinions of agents not adjacent to agent i in X i are assigned values such that
eT c i P Tx i ?0M and ?N 2,
then the resulting agreement state satisfies
the
global
constraint ,
(D T P )X c ?1 b .
4.2D pendulum dynamics and control
To investigate the viability of this approach,we apply the algorithm to the synchronisation of a distributed network of planar mass-cart pendula.The planar dynamics of the system allow us to analyse a network that is modelled by a simple form of the tree graph,a line graph.
4.1Dynamics
The dynamics of a single cart-pendulum system (referred to as an agent)can be derived using Lagrange’s equations (e.g.Spong 1989)
d d t @L @_q à
@L
@q ?Q ,L ?K àT ,where K is the kinetic energy of the system,T is the
potential energy of the pendulum,Q is the param-etrised forces acting on the system and L is the Lagrangian,where q is the vector of parameters,[v ]T as shown in Figure
5.
Figure 4.Directed tree graph example for initial conditions propagation.The figure represents the view of the graph from the perspective of agent X.Agent X assigns state opinions for non-adjacent agents according to the colours/hatchings shown in this diagram:in agents X’s state opinion vector,values of non-adjacent agents are set to values of agents adjacent to agent X that are in the path from the non-adjacent agent to agent
X.
Figure 5.Pendulum diagram.
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We can define the kinetic and potential energy as well as the parametrised forces acting on the system.Based on Figure 5,the only parametrised force,Q ,on the system is the force,F ,applied in the v direction.This force will be the control input,u ,to the system.No damping force is considered in this model as pendula can be approximated as zero damping sys-tems.The resulting equations of motion are
€
?à€v l cos e Tàg l sin e T€v
?àml € M tm cos e Ttml _ 2M tm
sin e Ttu
M tm :4.2Linearisation
These dynamics can be linearised about the
?0,_
?0,_v ?0equilibrium point.Here,we denote v i and i as the velocity and angle associated with the i th pendulum.As such,the single pendulum system for the i th pendulum is given as _x
ii et T?A i x ii et TtB i u i et T,where:x ii ?v i
_v i i _ i
2666437775,A i ?01000000000100àeM tm Tg
Ml
0266664377775
,B i ?01M 0à1Ml 26666643777775:
Note that this pair,(A i ,B i ),is completely controllable.
For an N planar pendulum system,the system can
be written as _x et T?Ax et TtBu et T,where x T ?x T 11,...,x T NN
??,u ?u 1...u N ??0T
,A ?A 1
0 0
A N
264375,B ?B 10..
.0
B N
2643
7
5:Note that in this article,A i ?A j and B i ?B j for i ,j ?1,...,N ,since the pendula are assumed to be homogeneous.
4.3Assumptions
Throughout this section,some assumptions are made and here we gather the assumptions for the sake of easy reference.We first assume that each cart-pendulum
system can measure its own cart position,cart velocity,
pendulum angle and pendulum angular velocity.It is also assumed that pendulum angles and angular veloc-ities are small enough so that the linearised dynamics can be used to model the system behaviour.Damping is also assumed to be small enough to approximate it as exerting zero forces on the system.It is also assumed adjacent agents can communicate state opinion vectors with each other.
4.4Planar pendula network
Since the pendula lie on a plane,the natural resulting network topology is a line graph as shown in Figure 6.
The corresponding pairwise constraint is that the adjacent pendula achieve synchronisation or,more specifically,that they achieve identical angles,angular velocities and cart velocities while maintaining a set distance,d ,between the carts.In terms of the pendula
states,it is required that v i àv j ?d ,_v
i à_v j ?0, i à j ?0,_
i à_ j ?0,i.e.P (x i àx j )?b where P ?I 4and b ?[d 000]T .
With the network topology and pairwise con-straints defined,each agent must define its state opinion vector.In addition,the global constraint can be defined using (4)and it is clear that we wish to drive the system to a state that satisfies this constraint.What must now be defined is how the system will be driven to such a state,once it is found,and how the state opinion vectors will be initialised for both adjacent and non-adjacent agents.The following constrained optimal control problem will give us both a way to initialise the state opinion vectors for each agent as well as a control law to drive each pendula to the agreement state that results from the consensus equation.
4.5Optimal control
For the following control law derivation,we treat a collection of pairwise constraints as a terminal con-straint on the linearised N agent system.This linear state constraint is denoted by Cx (T )?k .
The associated minimum energy,point-to-point transfer control problem with terminal time,T ,becomes
min u
J eu et TT?
Z T
jj u et Tjj 2d t e24
TFigure 6.N pendula line graph.
8R.Chipalkatty and M.Egerstedt
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such that
_x et T?Ax et TtBu et T,x e0T?x 0,x eT T?x T ,
where we assume that x T satisfies the constraints,i.e.
Cx T ?k .The solution to this common optimal control problem is
u opt ex T T?B T e A
T
eT àt T
W à1ex T àe AT x 0T,
where the Grammian,W ,is invertible and positive definite due to the controllability of the system.Plugging u opt (x T )back into the cost gives J (u opt (x T ))?
?ee AT x 0àx T TT e àA T
T W à1e àAT ee AT x 0àx T T:
Since x T is not unique,the goal now is to find the x T that minimises (25),which can be formulated as a quadratic programming problem,
min x T 1
2x T
T
Qx T tRx T such that Cx T ?k ,where
Q ?2e àA T
T W à1e àAT ,
R ?à2x T 0W à1e
àAT
:The unique solution,
x T opt ?Q à1eàR T tC T eCQ à1C T Tà1ek tCQ à1R T TT,
e25T
gives the control,
u opt ex T opt T?B T e A
T
eT àt T
W à1ex T opt àe AT x 0T:
e26T
The enabling observation now is that we can use
(25)to initialise the state opinions of adjacent agents as to where they should be driven such that the local constraints are satisfied and subsequently use these state opinions to initialise state opinions of non-adjacent agents.Then,the consensus equation can be utilised to update each pendulum’s state opinion vectors.As shown earlier,a final state opinion vector will be found that satisfies the global constraint for the entire network and the control law,(26),can be used to drive the system to this state,synchronising the pendula.
4.6System control:consensus algorithm Y optimal
control
The following summarises the algorithm used.First,each agent communicates its current state to its neighbours.Then,using these states,each agent
calculates its initial state opinion for its neighbours
using (25).Following this,each agent then assigns state opinions to non-adjacent agents using (16).These values are now collected in a state opinion vector for each agent.Next,each agent will communicate its state opinion vector at each time instant and use the consensus equation to update its state opinion vector.Finally,the state opinion vector is then used in (26)to calculate a control input.The state opinion vector update and control input calculation are repeated at every time instant until the terminal time,T .
Note that this implies that the consensus equation converges prior to the terminal time of the optimal control problem.For example,for an agent i not on the boundary (i.e.not agent 1or N ),the agent receives the state of agent i à1and agent i t1and creates the vector x (0)?[x i à1x i x i t1]T with
C ?P àP 00P àP !e27T
and k ?[b T b T ]T .These quantities are then used in (25)to initialise the state opinions of the agents two neighbours.After updating the state opinion vector through the consensus equation,agent i uses the values X i (t )to calculate its control at time t ,for i ?1,...,N .Let X ii (t )be the i th element of X i (t )such that the following control law is calculated,
u i eX ii et TT?B T i e
A T
i eT àt T
W i et Tà1eX ii et Tàe A i T x ii et TT:e28TRecall x ii is the current state of agent i and A i and B i are the corresponding single pendulum state space model.W i (t )is the Grammian for the pair (A i ,B i )from time t to T .We now have a control that drives the entire network to a terminal state that satisfies the global constraint.
4.7Simulation results
The stated control laws are implemented in a MATLAB simulation of the presented pendulum dynamics.Simulations are run with the following parameters:g ?9.8m/s,l ?0.30m,M ?1kg,m ?0.2kg,and d ?1.0m for the pendulum model.It should be noted that in order for this distributed control strategy to be effective,the consensus algo-rithm must converge to the agreement value before the specified final time in the optimal control law,which in this case is 20s.
In Figure 7,the results are shown for a five pendula scenario using the optimal control law and the consensus algorithm.As a comparison,the same initial conditions are run for the centralised case,where full network state information is known to all agents,i.e.only the optimal control law is needed without the
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consensus equation.The results of this case are shown in Figure 8.
It can be seen for both cases that at 20s,the distance between adjacent pendula is close to 1m,as prescribed,while the velocities,angles and angular velocities are identical for all the pendula.The anima-tions of these scenarios are given in Figure 9.
4.8Experimental results
Simulations of the proposed decentralised control system were deemed successful,so to further show the viability of this approach,a physical hardware demonstration was built.The testbed consists of three planar mass-cart pendula placed on three different parallel tracks as shown in Figure 12(b).The goal is to synchronise the pendula swinging with zero cart position and velocity after 10s starting from an
unsynchronised initial condition.The representative graph of the communication exchange and constraint directions is a 3node line tree graph in the same configuration as Figure 6with N ?3.
The demonstration structure and mass-cart pendula was constructed using Lego blocks as they serve as an effective rapid prototyping platform.Aluminium tubing was used as a track for the mass-cart pendula to slide upon and a Vex Sprocket chain was used to propel the carts.The effective pendulum length was 32.8cm.The chain sprockets were driven by AX12tServo motors as part of the Robotis Bioloid Robotics Kit.The Servo motors are connected to a CM-5micro-controller which receives commands over a serial communication port from a PC.The control software is implemented on the PC in a Java pro-gramming environment.
The AX12tServo motors are equipped with integrated encoders relaying position and velocity
5
10
15
20
25
1
2
3
4
5
6
7
Position vs Time
Time (s)P o s i t i o n (m
)
(a)
(b)(d)
(c)0
5
10
15
20
25
?0.2
?0.15?0.1
?0.05
00.050.1
0.15
0.20.25
Velocity vs Time
Time (s)V e l o c i t y (m /s
)
5
10
15
20
25
?1.5
?1
?0.5
0.51
1.5
Theta vs Time
Time (s)
T h e t a (r a d /s
)
5
10
15
20
25
?8
?6
?4
?2024
68Ang Velocity vs Time
Time (s)
A n g u l a r V e l o c i t y (r a d /s 2
)
Figure 7.Synchronisation control of 5distributed pendula results showing final positions are 1m apart with identical final velocity,angle and angular velocity.
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sensor information at 0.10s update rate with a 0.0051radian resolution.A Nubotrics WW-02WheelWatcher optical encoder was mounted to each pendulum to sense pendulum angle and angular velocity.Data was updated at a 0.1s rate to match the AX12tsensor rate at a 0.049radian resolution.This particular sensor was chosen for its floating optical wheel that minimises friction.
The system hardware/software architecture is shown in Figure 10.Motor commands are sent out as velocity commands as the motors cannot be controlled by force commands.Software for the CM-5is written in C code.Sensor data is passed through a fourth-order FIR low-pass filter designed in MATLAB.
The initial conditions for the demonstration run as seen in the attached video were created by running each pendulum at a sinusoid function of different phase shifts at the natural frequency of the pendulum,
0.8688Hz,for 5s.Then,the velocity control com-mands are then applied for 10s.As shown in Figure 11,at the end of the 10s.period,the pendulum angles and angular velocities are within 0.049radians of each other implying that the pendulum swinging is synchro-nised within that error.The data in the plots are smoothed using a fourth-order spline in MATLAB and the 0.049radian angle resolution error is compensated for in the pendulum angle data.
5.3D pendulum dynamics and optimal control We,now,want to show the versatility of this technique by applying it to a network system modelled by any arbitrary tree graph.The application is extended to the synchronisation of a distributed network of spherical pendula.We develop the application further by defying the dynamics,control and consensus problem
0246
810121416
?3.5
?3?2.5?2?1.5?1?0.500.51Position vs Time
Time (s)P o s i t i o n (m
)
(c)
(d)
(b)(a)
0246
810121416
?0.2
?0.15?0.1?0.0500.050.10.150.2
Velocity vs Time
Time (s)
V e l o c i t y (m /s
)
2
4
6
810
12
14
16
?1
?0.8?0.6?0.4?0.200.20.40.6
0.81Theta vs Time
Time (s)
T h e t a (r a d /s
)
0246810121416
?6
?4
?2
2
4
6
Ang Velocity vs Time
Time (s)
A n g u l a r V e l o c i t y (r a d /s 2
)
Figure 8.Synchronisation control of 5centralised pendula results showing final positions are 1m apart with identical final velocity,angle and angular velocity.
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associated with this example.The main differences from the planar example are the network topology and pendulum dynamics.The consensus algorithm and control laws are similar to the planar pendula case.
5.13D pendulum dynamics
The dynamics of a single cart-pendulum system (referred to as an agent)can be derived using Lagrange’s equations (e.g.Spong 1989).We can define the parameters and forces acting on the system through Figure 13(a).The only parametrised forces
?1.2
?1?0.8?0.6?0.4?0.20
Synchronising Pendulums
6
6.5
7
7.5
8
8.5
9
9.5
10
?1.2
?1?0.8?0.6?0.4?0.2
Synchronising Pendulums
(c)
(b)
(a)
?1.2
?1?0.8?0.6?0.4?0.20
Synchronising Pendulums
Figure 9.Animation of five pendula showing (a)initial conditions and final synchronisation for both (b)distributed and (c)centralised cases after 25
s.
Figure 10.System architecture of three pendula platform.
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(b)
(a)
Figure12.Demonstration platform images:(a)demonstration testbed,(b)Carts and(c)Servo motors.
051015
?0.2
?0.15
?0.1
?0.05
0.05
0.1
0.15
0.2
0.25
Angle vs Time
Time
A
n
g
l
e
(a)(b)
051015
?1.5
?1
?0.5
0.5
1
1.5
Anglular Velocity vs Time
Time
A
n
g
l
u
l
a
r
V
e
l
o
c
i
t
y
Figure11.Demonstration results showing synchronisation of experimental pendula:(a)pendulum angle and(b)pendulum
angular velocity.
International Journal of Control13 D
o
w
n
l
o
a
d
e
d
b
y
[
P
e
k
i
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a
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acting on the system,F v and F z ,applied in the v and z directions,will cause oscillations of the and angles.These forces will be the control inputs,u v and u z ,to the system.No damping force is considered in this model as pendula can be approximated as zero damping systems.However,these dynamics cannot be linearised about the ?0hanging equilibrium point because this point represents a singularity in the Jacobian of the system.As a result,Yang,Kuen,and Li (2000)propose an approximation of the dynamics given that the angle from vertical, stays under 10 .When this is the case,the pendulum can be projected onto the v –w and z –w planes such that new angles, and ,can be defined.This separates the system such that changes purely as a result of u z and as a result of u v .For small angles,this approximation assumes the projected pendulum length is l .Figure 13(b)shows the resulting system and the resulting equations of motion are
€v
?àml € M tm cos e Ttml _ 2M tm sin e Ttu v
M tm
,€z ?àml € M tm cos e Ttml _ 2M tm sin e Ttu z
M tm ,
€
?à€v
l
cos e Tàg l sin e T,€
?à€z l
cos e Tàg l sin e T:The resulting single pendulum system linearised about
the ?0,_
?0, ?0,_ ?0,_v ?0,_z ?0equilibrium point is _x
ii et T?A i x ii et TtB i u i et T,where x ii ?v i
_v
i z i _z i i _ i i _ i 266666666666643
777
77
7777
7
77
5
,B i ?00
1M 00001M 00à10000
à1Ml
2
66
666666666666664377777777777777775,A i ?010000000000000000010
00
0000000000
000
1000000
àeM tm Tg
Ml
000000000010
000
àeM tm Tg
Ml
2
66666666666666643
77777
77
7
777
7
77
7
5
:
Note that this pair,(A i ,B i ),is completely controllable.
For an N planar pendulum system,the system can
be written as _x et T?Ax et TtBu et T,where x et T?x T 11et T,...,x T NN
et T??T
,u et T?u 1et T,...,u N et T??T
,
A ?A 10 0
A N
264375,B ?B 10
..
.0
B N
2643
7
5:e29T
Recall that in this article,A i ?A j and B i ?B j for i ,j ?1,...,N ,since the pendula are assumed to be homogeneous.For this example,we will be using the
example graph,^G
,as a representation of the distrib-uted cart-pendula system.Given this system,a control law is sought to drive these pendula in such a way that they achieve identical angles,angular velocities and cart velocities while maintaining a set distance, ,between the carts.This requirement will be the pairwise constraints enforced between adjacent agents i and j such that (3)is enforced,for
P ?I 8?8??,e30T
b ? 1
0 2
00
0?
?T ,
e31T
with distances 12R 1in the v direction and 22R 1in the z direction such that ???????????????
21t 22q .I 8?8is the
identity matrix.It should be noted that can be different for each pairwise constraint on the graph.The assumptions made about this system are identical to the assumptions made in the planar pendula case in the last section.
5.2System control:consensus
algorithm Y optimal control Now that the system dynamics and pairwise con-straints are defined for a given network graph,we define the consensus problem to be the same as the planar case.The only difference
is that we are
now
Figure 13.Pendulum diagrams:(a)with hanging singularity
and (b)Modified with small angle approximation.
14R.Chipalkatty and M.Egerstedt
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dealing with a larger class of network topologies,tree graphs instead of just line graphs.Let d i indicate the
number of adjacent agents to agent i for all i 2^V
.Each pendulum can calculate state opinions for its neigh-bours based on its state,x ii (t )2R 8,and the state of its d i adjacent agents,x jj (t )2R 8for j 2N i .This local state opinion vector is defined as X d i et T??y 1,...,x i ,...,y d i T 2R 8?N ,where
y k ?x jj et Twhere j is the k th agent in N i ,è
e32Tfor k ?1,...,d i and j ?1,...,N .For agent i ,A d i and B d i are the corresponding pendula state space models for d i adjacent agents.W d i et Tis the Grammian for the pair eA d i ,B d i Tfrom time t to T .
Here,each pendulum can initially solve for a optimal terminal state based on the local state opinion vector that satisfies its local constraint at t ?0.Let i be the matrix T i with all columns with every element being zero removed.From Section 5,the optimal terminal state,used to initialise the state opinions for adjacent agents,is given by the unique solution,
x iT ?Q à1i eàR T i tC T i eC i Q à1i C T i Tà1ep i tC i Q à1i R T
i T,
e33T
where
Q i ?2e àA T
di T W à1di e àA di T
,e34TR i ?à2X T d i e0TW à1di e
àA di T ,e35TC i ? i P ,e36Tp i ?f i b :
e37T
The state opinion vector,x i ,for this problem will be defined using x iT for adjacent agents state opinions and the state assignments in (16)for non-adjacent agent state opinions.
Then,X i (0),the initial conditions of the consensus equation,are defined from x i and .After this,the consensus equation updates the state opinion vector,X i (t ),for each pendulum based on the state opinion vectors of its neighbours.The state opinions for agent i can be extracted from the state opinion vector,X i (t ),and used to calculate an optimal control law to drive agent i .After every update,the new consensus values of agent i are used in the optimal control presented in Section 4.5.The control law,at time t is given by (26)for i ?1,...,N and the control input for agent i is u i 2R 2.We now have a control that drives the entire network to a terminal state that satisfies the terminal constraint.
It should be noted that the pendula converge to a set distance apart and have equal angles and angular
velocities.They also have equal cart velocities;how-ever,these velocities are not guaranteed to be zero.It
should also be noted that in order for this distributed control strategy to be effective,the consensus algo-rithm must converge to the agreement value before the specified final time in the optimal control law,T .
5.3Simulation results
The stated control laws are implemented in a MATLAB simulation of the presented pendulum dynamics.Simulations are run with the following parameters:g ?9.8m/s,l ?0.30m,M ?1kg,and m ?0.2kg for the pendulum model.For each edge,a different was assigned:edge 1( 1?1.0m, 2?2.0m),edge 2( 1?à5.0m, 2?6.0m),edge 3( 1?3.0m, 2?1.0m)and edge 4( 1?à4.0m, 2?3.0m).It should be noted that in order for this distributed control strategy to be effective,the consensus algo-rithm must converge to the agreement value before the specified final time in the optimal control law,which in this case is 20s.
In Figures 14and 15,the results are shown for a five pendula scenario using the optimal control law and the consensus algorithm.
It can be seen that at 20s,the distance between adjacent pendula is close to the respective ’s,as prescribed,while the velocities,angles and angular velocities are identical for all the pendula.The anima-tions of these scenarios are given in Figures 16(a)and 16(b).
6.Conclusions
The main contribution of this article is a novel method for reaching agreement among distributed agents (organised in a tree graph topology)such that a globally defined constraint is satisfied over the network given only local information.This method only requires a priori knowledge of the network topology and a static update law that only needs to be initialised and updated using local information.The presented work advances the state of the art in that previous work on constrained agreement has required time-varying update laws where weights used in the consensus equation must be recomputed for every iteration.
In other words,one just needs to set-up each agent’s initial state opinion vector,which can be done using only information communicated by its neigh-bours,and then simply run the standard consensus equation over the network.The result of the consensus equation is a network state that satisfies the global constraint.
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0102030?0.5
0.5Theta vs Time
Time (s)T h e t a (r a d /s
)
0102030
?2
?1012Theta Ang Velocity vs Time
Time (s)
A n g u l a r V e l o c i t y (r a d /s 2
)
0102030?0.4
?0.200.20.4Phi vs Time
Time (s)
P h i (r a d /s
)
0102030
?2
?1012Phi Ang Velocity vs Time
Time (s)
A n g u l a r V e l o c i t y (r a d /s 2)
Figure 15.Five pendulum simulation results showing identical angles and angular velocity after the 20s mark showing pendula synchronisation.
0102030
?5
5
10
X Position vs Time
Time (s)
P o s i t i o n (m
)
0102030
?0.5
0.5X Velocity vs Time
Time (s)
V e l o c i t y (m /s
)
0102030
?5
0510
Z Position vs Time
Time (s)
P o s i t i o n (m
)
0102030
?0.5
0.5
1
Z Velocity vs Time
Time (s)
V e l o c i t y (m /s
)
Figure 14.Five pendulum simulation results showing position formation and identical velocity.X and Z positions plots show that the adjacent agents maintain inter-agent distance while the velocity plots show identical velocities after the 20s mark.
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This technique was applied to the control of a distributed network of linearised planar and spher-ical pendula.This algorithm,together with point-to-point transfer optimal control,was able to drive these linear systems to a terminal manifold such that the manifold synchronises the pendula oscillations and maintains a desired cart formation.
Future research directions include relaxing the requirements for controllable linear systems as well as the need for tree graph topologies.Specifically,the point-to-point transfer optimal control would have to be reformulated and solved for using nonlinear dynamics.In addition,uncontrollable systems would limit the states each system could reach;hence,constraints on the final agreement state must be added that require it to lie in the controllable subspace of every agent in the network while also satisfying the global constraint.Applying the method to topologies other than tree graphs requires the development of an alternate method for initial condition assignment because cycles make the assignment propagation an overdetermined problem (i.e.more than one agent has an opinion on what the unknown value should be).
Acknowledgements
This work was supported by the US National Science Foundation through Grant #0820004.The authors would also like to thank Dr Patrick Martin for the use of the software platform underpinning the experimental platform.
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?3
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Figure 16.Five pendulum simulation showing:(a)initial conditions and (b)synchronisation after 25s.
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18R.Chipalkatty and M.Egerstedt
D o w n l o a d e d b y [P e k i n g U n i v e r s i t y ] a t 08:13 21 M a r c h 2012
一年级数学分析报告 一、考试情况概述： 一年级数学分析报告.整个试卷注重了基础知识的训练，体现“数学即生活”的理念，让学生用学到的数学知识，去解决生活中的各种数学问题.本次质量检测试卷注重人文性，体现数学与生活的实际联系.避免非知识非智慧非数学错误的产生，还体现在试卷的图文并茂、生动活泼，给学生以亲切感. 二、对试卷习题及考试成绩的分析： 本次期中质量检测，应考人数5人，实考人数6人.100-95分5人，1人是93分.平均成绩达到93分，及格率100%，优秀率100%.从学生做题情况来看，学生的基础知识掌握的比较好，基本功扎实，形成了一定的基本技能. 试卷难易程度总体适中，基础题考核全面，呈现的基础性强，后部分应用题，稍有偏难，一年级部分学生达不到那种审题能力.从本次考试的难易程度和所取得的成绩来看，大部分小朋友都有所进步. 第一题，请同学们先数一数，再连一连.共有9/3/4/2/8等5个图形和5个数字，让学生看图数数，再和相应的数字连线.由于对此类题型教师注重训练，学生基础知识扎实，准确率达100%. 第二题，我会看图写数，考查学生对图形的认识和能否数出图形中物体的个数，并写数，此题图文并茂，通俗易懂，准确率达到100%. 第三题，想一想，画一画，包括三个小题.考查学生的逻辑思维能力,对前后、左右和数序这个知识点的掌握情况，多数学生都对，个别学生由于审
题不认真，出现错误.在今后的教学中要加强培养学生读题和审题的能力训练. 第四题，我会比较.考查学生会用认识大于、小于、等于这个知识点的掌握情况.个别学生认识把大于误写成了小于出现错误. 第五题，看一看、数一数、填一填，要求学生根据小狗、小兔、小牛、小象四个小动物的排列顺序，让学生掌握数列和数序，学生答题的准确率是100%. 第六题，在方框里填上合适的数，让学生根据数的组成，数的分解完成练习，大部分学生都对，也有个别学生出现错误. 第七题，画一画，比一比，学生通过动手操作，画图并比较，培养学生的动手操作能力和逻辑思维能力，通过画图和比较，学生知道了多几个和少几个的关系，能够正确的进行解答. 第八题，比一比、填一填，学生通过比较两种图形的多和少，知道哪个比哪个多，哪个比哪个少的关系，能够用数学解决问题. 第九题，看谁算的又对又快，本题是一年级数学教学的重点，我在平时的教学中，注重培养学生的计算方法和计算能力，学生在答卷时速度快、准确率高，失分较少. 第十题，数一数、填一填，本题是让学生分别数出长方体、圆柱、正方体、球各有几个，在相应的括号里填数，准确率达到100%. 第十一题，根据图意，列算式并计算，共6个小题，实际上是要求学生看图列式，要求学生仔细观察图，明白图意，然后根据图意列式计算.由于图意简单醒目，前四小题学生看图后就能理解，所以全对.最后两个小题，
一年级数学试卷分析报告新版 学期已结束了,我以诚恳的工作态度完成了期末的数学检测工作.现将年级本期的数学检测卷面评析简析如下： 一.基本情况 本套数学试卷题型多样,内容覆盖面广,题量恰当,对于本学期所学知识点均有安排,而且抓住了重点.本次期末考试共有39人参加, 及格率92.11％,优秀率92.11％,全班最高分100分,平均分82.92分. 二.学生答题分析 1.学生答题的总体情况 对学生的成绩统计过程中,大部分学生基础知识扎实,学习效果 较好,特别是在计算部分.图形的认识,这部分丢分较少.同时,从学生的答卷中也反映出了教学中存在的问题,如何让学生学会提出问题. 分析问题.并解决问题,如何让我们的教育教学走上良性轨道,应当引起重视.从他们的差异性来分析,班级学生整体差距比较大的,说明同学之间还存在较大的差距,如何扎实做好培优辅差工作,如何加强班 级管理,提高学习风气,在今后教育教学工作中应该引起足够的重视.本次检测结合试卷剖析,学生主要存在以下几个方面的普遍错误类型：
第一.不良习惯造成错误.学生在答题过程中,认为试题简单,而产生麻痹思想,结果造成抄写数字错误.加减号看错等. 第二.审题不认真造成错误.学生在答题过程中,审题存在较大的问题,有的题目需要学生在审题时必须注意力集中才能找出问题,但学生经常大意. 2.典型题情况分析 （1）填空题：学生对填数和数物体掌握较好,但在第4小题找规律填数.第7小题59.90元表示（）元（）角这几道题失分较多,学生在理解59.90元表示什么的这方面还有一定的困难. （2）算一算：有20以内的退位减法.两位数加减整十数.两位数加.减一位数（进位加.减）,还有小括号的认识,这部分计算学生能够有效掌握计算方法,总体失分在2分左右,一小部分同学在这一块失分主要是马虎大意,看错+.-符号,另外还有个别同学在计算技能上稍有欠缺. （3）比一比:主要是考查两位数比较大小,此外还对人民币的认识知识略有涉及,考查了人民币单位换算及大小比较,学生基本上都能够正确解答,这部分失分较少. （4）选一选：在合适的答案下面打“√”,这一题考查学生对“多一些”“多得多”“少一些”“少得多”之间的理解,试卷上出现“接
2016小学一年级数学下册期末试卷质量分析报告 2016小学一年级数学下册期末试卷质量分析报告 一年级数学期末试卷分析 一、试题情况分析 数学题量适中，题型多样化，知识面广，接近生活，尤其是个别试题特别灵活，可多种方法操作，适合中上等生。偏难。 二、错题情况分析 第一题，填空 （1）“一个两位数，十位上的数比个位上的数小4，这个数可能是、。”这道题错的特别多。大部分同学不理解题意，不知从何入手。 （2）“把58、79、12、33、95、80从小到大排列。”数字多，括号小，同学们答的乱。 （3）“正方形的四条边，长方形的对边。”同学们看图形能分辨两种图形，但是叙述图形特点不准确。对于这类题不理解，不知怎样表达。 （4）“3张1元，2张5角，5张1角组成。”这道题数字多，学生思考不能周全，大部分学生心里糊涂，不知是多少钱了。 第二题，写出钟面表示的时刻。也有很多学生错。有的是学生答题不认真，有的是学生没有答题方法。 第三题，计算。加减混合错的多。大部分学生是马虎，个别学生是基础差，根本就不会算。 第四天，填符号“25角2元6分”由于“分”这个单位在生活中不常用，孩子接触少，有的甚至没有“分”
这个概念，不知“分”到底有多大，所以不会比较。“34=81”思维不灵活的学生符号填错。 第五题解决问题，条件，问题对于一年级下等孩子来说给的太多，孩子们的思维还达不到这个清晰度，有的计算出现错误。还有就是提出问题，许多孩子提出问题的语言不准确，没有标点符号。平时训练不到位。 第六题，统计的第5小题错的多，有的学生没读通题，只提问，不解答。 第七题，画一画的第2小题，画四种不同的图形，属于奥数问题，对于中下等生来说偏难，不会全面思维。 三、改进措施 （1）要加强基础知识的教学，培养学生灵活计算的能力。 （2）平时要注意引导孩子知识与生活实际相结合，让知识给生活服务。 （3）培养孩子的创新思维，一题多解或一题多变，能从不同角度把握知识、运用知识。 （4）注重孩子学习习惯的培养，说完整话，书写工整。 绿色圃中小学教育网免费提供新课标人教版北师大版苏教版试卷课件教案作文，右边中间有分享按钮，喜欢本站，请点右边分享到自己的博客空间微博等。 一、试题整体情况： 本次期末考试试卷从总体来看试卷抓住了本年级本册书的重点、难点、关键点。整个试卷注重了基础知识的训练，体现"数学即生活"的理念，让学生用学到的数学知识，去解决生活中的各种数学问题。 本次试卷共有十二大题，不仅考查了学生对基本知识的掌握，而且考查了学生的数学学习技能，还对数学思想进行了渗透。 二、学生答题情况：
一年级数学试卷分析报 告新编 TYYGROUP system office room 【TYYUA16H-TYY-TYYYUA8Q8-
何寨中心小学一年级数学期末试卷分析学期已结束了，我以诚恳的工作态度完成了期末的数学检测工作。现将年级本期的数学检测卷面评析简析如下： 一、基本情况 本套数学试卷题型多样，内容覆盖面广，题量恰当，对于本学期所学知识点均有安排，而且抓住了重点。本次期末考试共有39人参加,及格率％,优秀率％,全班最高分100分，平均分分。 二、学生答题分析 1、学生答题的总体情况 对学生的成绩统计过程中，大部分学生基础知识扎实，学习效果较好，特别是在计算部分、图形的认识,这部分丢分较少。同时，从学生的答卷中也反映出了教学中存在的问题，如何让学生学会提出问题、分析问题、并解决问题，如何让我们的教育教学走上良性轨道，应当引起重视。从他们的差异性来分析，班级学生整体差距比较大的，说明同学之间还存在较大的差距，如何扎实做好培优辅差工作，如何加强班级管理，提高学习风气，在今后教育教学工作中应该引起足够的重视。本次检测结合试卷剖析，学生主要存在以下几个方面的普遍错误类型：第一、不良习惯造成错误。学生在答题过程中，认为试题简单，而产生麻痹思想，结果造成抄写数字错误、加减号看错等。 第二、审题不认真造成错误。学生在答题过程中，审题存在较大的问题，有的题目需要学生在审题时必须注意力集中才能找出问题，但学生经常大意。 2、典型题情况分析
（1）填空题：学生对填数和数物体掌握较好，但在第4小题找规律填数、第7小题元表示（）元（）角这几道题失分较多，学生在理解元表示什么的这方面还有一定的困难。 （2）算一算：有20以内的退位减法、两位数加减整十数、两位数加、减一位数（进位加、减），还有小括号的认识，这部分计算学生能够有效掌握计算方法，总体失分在2分左右，一小部分同学在这一块失分主要是马虎大意，看错+、-符号，另外还有个别同学在计算技能上稍有欠缺。 （3）比一比:主要是考查两位数比较大小，此外还对人民币的认识知识略有涉及，考查了人民币单位换算及大小比较，学生基本上都能够正确解答，这部分失分较少。 （4）选一选：在合适的答案下面打“√”，这一题考查学生对“多一些”“多得多”“少一些”“少得多”之间的理解，试卷上出现“接近”这个词语时，部分同学不能够理解这个词语的意思，导致失分，看来学生思维还不够灵活，平时还应做到举一反三。 （5）做一做：这部分有5道小题，考查学生的解决问题的能力。第1小题帮妈妈购物，学生失分较多的在④题，在理解题目意思上还有一定的困难。第2、3题看图列式，第5题解决问题，这3道题考查两位数加两位数进位加属于二年级学习的内容，导致学生失分较多。 三、问题与分析 （一）存在问题 根据以上分析，主要存在的问题有： 1．学生整体观察题目的意识和习惯不够，对题的特征缺乏敏感性。
福建省高等教育自学考试应用心理学专业（独立本科段） 《心理学研究方法》课程考试大纲 第一部分课程性质与目标《心理学研究方法》是福建省高等教育自学考试应用心理学专业（独立本科段）的一门专业基础必修课程，目的在于帮助考生了解和掌握心理学研究的理论基础和主要方法，检验考生对心理学研究理论基础与主要方法，检验考生对心理学研究方法的基本知识和主要内容的掌握水平与应用能力。 心理学研究的对象是心理现象。它的研究主题十分广泛：即涉及人的心理也涉及动物的心理；即涉及个体的心理也涉及群体的心理；即涉及有意识的心理也涉及潜意识的心理；即涉及与生理过程密切相关的心理也涉及与社会文化密切相关的心理。心理学研究是一种以经验的方式对心理现象进行科学探究的活动。由于心理学的研究方法是以经验的或实证的资料为依据的，因而使心理学与哲学相区别，也与人文学科相区别。 设置本课程的具体目的要求是，学习和掌握心理学研究方法的基本理论和基本技能，将有助于学生们理解心理学的基本概念、基本原理和基本理论。理解心理学家在探索心理与行动时所做的一切，有助于考生将来为心理学的发展做出有益的贡献。 第二部分考核内容与考核目标 第一编心理学研究基础 第一章心理学与科学 一、学习目的与要求 通过本章学习，要求考生了解心理学的性质，了解心理学科学研究的方法、特征及基本步骤，理解心理学研究的伦理问题和伦理规范。 二、考核知识点与考核目标 1、识记： （1）心理学的含义； （2）心理学科学研究的特征：系统性、重复性、可证伪性和开放性； （3）知情同意。 2、领会： （1）一般人探索世界的常用方法； （2）心理学研究主要包含哪几个步骤； （3）科学研究的开放性主要表现在哪几方面； 3、应用： （1）根据科学研究的特征来分析某些心理学的研究； （2）心理学研究的伦理问题及以人为被试的研究的伦理规范来分析是否可以在心理学研究中使用欺骗的方法。
小学一年级数学上册期末质量分析报告 本次期末考试试卷从总体来说抓住了本年级本册书的重点、难点、关键点。整个试卷注重了基础知识的训练，体现“数学即生活” 的理念，让学生用学到的数学知识，去解决生活中的各种数学问题。本次试卷不仅考查了学生对基本知识的掌握，而且考查了学生的数学学习技能。从学生做题情况来看，学生的基础知识掌握的还可以，但基本功不是十分扎实，如书写水平，计算能力等基本技能还有待进一步提高，灵活解决数学问题的能力也有待进一步的提高。 学生各大题得、失分及成因情况分析 一、算一算，你真棒，四、投篮（连一连）： 这部分主要查学生的计算掌握情况，有一部分学生粗心大意，有一部分学生的计算能力较弱，没能熟练掌握计算技巧，因此没有百分百全对。 二、填一填，你能行： 第 1 小题，考查了数的认识，数的顺序，要求学生能按照数的顺序填数，中下学生对相邻两数不是相差 1 的数列掌握得不熟练，错误率较高，是整张试卷中失分最多的地方之一。 第 2、 3 小题考查了数的组成和比较数的大小，这道题学生得分率较高。但第三小题“8 和 3 的和是（），差是多少
（）” ，在上册还没有提到“和”和“差” 这两个概念，学生有一定的失分率。 第 4 小题，考察学生对“﹤，﹥，﹦” 的认识和使用，大部分学生能掌握这三个符号的使用，得分较高。 第 5 小题（） +5=12，考察学生对加减法逆运算的认识和掌握情况，中下生对这部分知识掌握较差。 第 6 小题，出现一组数，考察学生对位置与顺序的分辨水平，学生能较好地完成，但部分学生做题不认真，在给数字排序的时候没有按照要求的“从大到小” 的顺序排列，而是“从小到大” ，有不必要的失分。 三、照样子写时间。 调整后的第一册数学在认识时钟上只要求学生会辨认整时和半时，减少了认识快几时了和几时刚过，减轻了学生的负担。因此大部分学生懂得根据时针和分针的位置顺利写出整时和半时，但有些学困生对半时的认识一知半解，作答情况不好。 四、把不同类的圈起来： 该题是整张考卷中失分最严重的地方，有两个小题，第一小题是文具和玩具，应将铅笔圈出来，但学生审题不认真，把他们看成了交通工具和动物，把玩具鱼错圈。第二小题是平面图形和几何图形，应将平面图形三角形圈出来，但学生没有这方面知识的认识，平时教学也没有涉及这方面的知识，因此，学生将这些图形分为有角的和没有角的，将立体图形的球体错圈。
一年级数学期末测试质量分析报告 万家镇宅科小学 一、试题分析 本次试卷具有以下几项特点： 1、适合新课标理念，难易程度适中，内容全面，注重能力培养。 2、考核学生的基础知识、基本技能的同时，注重了对学生综合能力的考查。 3、题目注重对学生双向思维的考核，有利于学生思维的灵活性和创造 二、学生错误分析 结合试卷分析，我班学生答题主要存在以下几个方面的普遍错误类型。 1、不良习惯造成错误 学生在答题过程中，认为试题简单，而产生麻痹思想，结果造成抄写数字错误。 2、审题不认真造成错误 学生在答题过程中，审题存在较大的问题，有的题目需要学生在审题时必须注意力集中才能找出问题，但学生经常大意。例如：第一题第9小题，5依次往后算，学生没有仔细看题。 3、收集、处理信息和分析、解决问题的能力不强造成错误。 例如第三题（2小题）学生数圆出现问题特多。第六题（2小题）让学生提出问题并回答，属于开放性的题目，教材中有类似的练习，这里只是改变一下内容的呈现方式，目的是考查学生分析解决实际问题的能力，但学生失分严重，可见学生理解分析问题的能力较差。 三、对今后教学改进意见 1、注重良好习惯的培养。 从卷面上看，学生的审题不够认真，抄错数字，看错题目要求，计算粗心马虎等，是导致失分的一个重要原因。这些是长期不良习惯养成的后果，应当引起高度重视。其实养成良好的学习习惯，也是学生的一个基本的素质，它将使学生受益终生。 ２、注重开放题教学，引导学生在创新中学习。 小学数学开放题，因其开放性、多变性、灵活性给学生的思维创设了一个广阔的空间，有助于激发学生创新意识，养成创新习惯，发展思维的创造性，提高学生的实践能力。平时除了教学书本上的基础知识外，还要注意开放性题目的设计和训练，为不同层次的学生学好数学提供机会，不断实现学生创新能力与实践水平的发展。
一年级下册数学检测质量分析报告 在2019-2020学年度下学期一年级数学期末检测中，我班有xx 人参加考试，参考率为100%，总分是xxxx分，平均分是xxxx分.及格人数为xx人，及格率100%，优秀人数为xx人，优秀率xxxx。100分有x人，90~99有xx人，80~89有x人。本次期末检测体现课程理念，注重考查学生能力，对学生一学期的学习情况作了客观、公正的评价，从卷面的得分来看，总体成绩还不错，但还存在很多不足。为进一步加强教育教学管理，端正工作态度，严格工作作风，全面提高教育教学质量，现结合我班实际，整改报告如下： 一、试卷分析 本次试卷考核知识面广，题型灵活多变，既重视对学生基础知识的掌握，又注重让学生将所学的知识解决生活中的实际问题，充分体现新课标的要求。此次一年级数学试卷题型有填空题、判断题、画图题、计算题、应用题分别对学生的计算能力、概念的理解能力、运用能力、解决有关数学问题能力、对基本数量关系的分析能力、整理数据、处理数据的能力等多方面能力进行全面考查。 二、存在问题 1、个别学生计算不细心，没有养成良好的检查习惯，缺乏检验意识，看错、抄错题目、运算符号的低级错误现象时有发生。 2、学生读题、审题、分析问题和解决问题的能力比较差，因而在应用知识解决问题这部分知识中失分比较多。学生审题不严谨。部
分学生缺乏认真仔细的审题习惯，凭主观意愿来解题。这种情况反应出学生思维停留在表面，缺乏深度。因而灵活解题及认真审题的能力有待进一步提高。 3、综合运用知识的能力较弱。表现在学生填空、判断，主要原因学生在学习过程中对于新知识体验不深，头脑中建立的概念不清晰、不扎实。 三、剖析主观原因 1、在课堂教学中，学生的学习主动性欠缺，学生听课被动，教师关注学生教学任务较多，课堂管理较少，师生互动不够，中下游学生积极性没有调动起来。 2、自己的课堂教学，传统的“填鸭式”教学最显著，没有很好的调动学生的学习积极性和学习兴趣。评价性语言乏味。兴趣是最好的老师。如何使学生对学习产生兴趣，重点在于教师如何把课上活。上活课程最重要的思想就是“变通”。“变通”，简单理解就是变化了才通畅。课堂教学过程中，学生肯定会碰到许多不懂的或难以理解的内容，教师要善于变通，由浅入深的讲解，举一反三的例证，旁敲侧击的提示都行。只要你善变，敢变，必然会找到一条通路。 3、长期以来，由于应试教育的影响，数学作业内容拘泥于课堂知识，拘泥于教材，往往以试卷中出现的形式作为作业的模式，完成同步练习，机械、重复的较多。作业陷入机械抄记、单调封闭的误区不能自拔。那些限于室内，拘于书本的静态作业使学生埋头于繁琐重复的书面练习而苦不堪言。学生仍然停留在以“练”为主的机械式的
一年级数学试卷分析报告 This manuscript was revised by the office on December 10, 2020.
何寨中心小学一年级数学期末试卷分析 学期已结束了，我以诚恳的工作态度完成了期末的数学检测工作。现将年级本期的数学检测卷面评析简析如下： 一、基本情况 本套数学试卷题型多样，内容覆盖面广，题量恰当，对于本学期所学知识点均有安排，而且抓住了重点。本次期末考试共有39人参加,及格率％,优秀率％,全班最高分100分，平均分分。 二、学生答题分析 1、学生答题的总体情况 对学生的成绩统计过程中，大部分学生基础知识扎实，学习效果较好，特别是在计算部分、图形的认识,这部分丢分较少。同时，从学生的答卷中也反映出了教学中存在的问题，如何让学生学会提出问题、分析问题、并解决问题，如何让我们的教育教学走上良性轨道，应当引起重视。从他们的差异性来分析，班级学生整体差距比较大的，说明同学之间还存在较大的差距，如何扎实做好培优辅差工作，如何加强班级管理，提高学习风气，在今后教育教学工作中应该引起足够的重视。本次检测结合试卷剖析，学生主要存在以下几个方面的普遍错误类型： 第一、不良习惯造成错误。学生在答题过程中，认为试题简单，而产生麻痹思想，结果造成抄写数字错误、加减号看错等。
第二、审题不认真造成错误。学生在答题过程中，审题存在较大的问题，有的题目需要学生在审题时必须注意力集中才能找出问题，但学生经常大意。 2、典型题情况分析 （1）填空题：学生对填数和数物体掌握较好，但在第4小题找规律填数、第7小题元表示（）元（）角这几道题失分较多，学生在理解元表示什么的这方面还有一定的困难。 （2）算一算：有20以内的退位减法、两位数加减整十数、两位数加、减一位数（进位加、减），还有小括号的认识，这部分计算学生能够有效掌握计算方法，总体失分在2分左右，一小部分同学在这一块失分主要是马虎大意，看错+、-符号，另外还有个别同学在计算技能上稍有欠缺。 （3）比一比:主要是考查两位数比较大小，此外还对人民币的认识知识略有涉及，考查了人民币单位换算及大小比较，学生基本上都能够正确解答，这部分失分较少。 （4）选一选：在合适的答案下面打“√”，这一题考查学生对“多一些”“多得多”“少一些”“少得多”之间的理解，试卷上出现“接近”这个词语时，部分同学不能够理解这个词语的意思，导致失分，看来学生思维还不够灵活，平时还应做到举一反三。 （5）做一做：这部分有5道小题，考查学生的解决问题的能力。第1小题帮妈妈购物，学生失分较多的在④题，在理解题目意思上还有一定的困难。第2、3题看图列式，第5题解决问题，这3
心理学研究方法复习题 一、重要概念 1、研究的效度：即有效性，它是指测量工具或手段能够准确测出所需测量的心理特质的程度。 2、内部一致性信度：主要反映的是测验内部题目之间的信度关系，考察测验的各个题目是否测量了 相同的内容或特质。内部一致性信度又分为分半信度和同质性信度。 3、外推效度：实验研究的结果能被概括到实验情景条件以外的程度。 4、半结构访谈：半结构化访谈指按照一个粗线条式的访谈提纲而进行的非正式的访谈。该方法对访谈 对象的条件、所要询问的问题等只有一个粗略的基本要求，访谈者可以根据访谈时的实际情况灵活地做出必要的调整，至于提问的方式和顺序、访谈对象回答的方式、访谈记录的方式和访谈的时间、地点等没有具体的要求，由访谈者根据根据情况灵活处理。 5、混淆变量：如果应该控制的变量没有控制好，那么，它就会造成因变量的变化，这时，研究者选定 的自变量与一些没有控制好的因素共同造成了因变量的变化，这种情况就称为自变量混 淆。 6、被试内设计：每个被试接受接受自变量的所有情况的处理。 7、客观性原则：是指研究者对待客观事实要采取实事求是的态度，既不能歪曲事实，也不能主观臆断。 8、统计回归效应：在第一次测试较差的学生可能在第二次测试时表现好些,而第一次表现好的学生则 可能相反,这种情形称为统计回归效应.。统计回归效应的真正原因就是偶然因素变化导致的随机误差，以及仅仅根据一次测试结果划分高分组和低分组。 9、主体引发变量：研究对象本身的特征在研究过程中所引起的变量。 11、研究的信度：测量结果的稳定性程度。换句话说，若能用同一测量工具反复测量某人的同一种心理特质，则其多次测量的结果间的一致性程度叫信度，有时也叫测量的可靠性。 12、分层随机抽样：它是先将总体各单位按一定标准分成各种类型（或层）；然后根据各类型单位数与总体单位数的比例，确定从各类型中抽取样本单位的数量；最后，按照随机原则从各类型中抽取样本。13、研究的生态效度：生态效度就是实验的外部效度，指实验结果能够推论到样本的总体和其他同类现象中去的程度，即试验结果的普遍代表性和适用性。 14、结构访谈：又称为标准化访谈，指按照统一的设计要求，按照有一定结构的问卷而进行的比较正式的访谈，结构访谈对选择访谈对象的标准和方法、访谈中提出的问题、提问的方式和顺序、访谈者回答的方式等都有统一的要求。 15、被试间设计：要求每个被试者（组）只接受一个自变量处理，对另一被试者（组）进行另一种处理。
心理学研究方法复习资料 考试题型概念解释（5*4=20)、简答(3*8=24)、案例分析(4*10=40)、研究设计(1*16=16)。本资料包括老师提到的概念解释和简答，案例分析和研究设计来自于课本挑战性问题。本资料仅供参考。 名词解释: 1、研究假设（Hypothesis）：是由理论所推衍出来的更为具体的预测，是针对研究问题提出的有待验证的、暂时性的、推测性的解答。 2、实验混淆（confounding）：如果我们所设定的自变量其发生量的改变时，另一个已知或潜在的变量亦随之有量的改变，则这两个变量的作用就发生了相互混淆。 3、观察者间信度（interobserver reliability）：指不同的独立的观察者针对同一观察所做记录的一致性程度。 4、便利抽样（convenience sampling）：主要根据获得调查对象的容易程度和调查对象参加调查的意愿所进行的。便利抽样是所有取样技术中花费最小的，抽样单元是可以接近的、容易测量的，并且是可以合作的。便利取样是非随机抽样，所选取的样本缺乏代表性。 5、假相关（spurious relationship）：如果两个变量间的关系可以通过第三个变量进行解释，那么这种关系就被称之为“假相关”。 6、操作定义（operational definition）：指仅根据可观察的程度来解释概念，这个观察程度是来产生和测量概念的。即从具体的行为、特征、指标上对变量的操作进行描述，将抽象的概念转换成可观测、可检验的项目。 7、外部效度（external validity）：是指研究结果能够一般化和普遍化到样本来自的总体和其他条件、时间、和背景中去的程度。 8、内部效度（internal validity)：指实验中的自标量与因变量之间因果关系的明确程度。如果在试验中：当自变量发生变化时因变量随之发生改变，而自变量恒定时因变量则不发生变化，那么这个实验就具有较高的内部效度。内部效度与无关变量的控制有关。 9、交互作用（interaction effect）：是指一个因子的效应依赖于另一个因子的不同水平，当一个自变量的效应在另一自变量的不同水平上存在差异时，就表示出现了交互作用。10、知情同意(informed consent)：是被试在充分理解研究性质、不参加的后果、影响参加意愿的所有因素后，明确表达参加研究的意愿。 11、统计效力（power）：指研究者在进行统计检验时，正确拒绝虚无假设的可能性，换言之，就是当存在真正效应时检验发现效应的概率。统计效力受到统计显著性水平、处理效应大小
工作总结 xx小学一年级xxxx 眨眼间一个学期就过去了。在这短短的一个学期里，我获益良多。 本学期,我担任一（1）班的数学教学工作，作为刚接一年级的数学教师，感受到自己肩上的担子很重。由于低年级数学教学经验尚浅，加之学生刚刚由幼儿园升入,学习习惯还未养成,再加上家庭及学生智力的差异，因此，我对教学工作不敢怠慢，认真学习，虚心向其他教师学习。立足现在，放眼未来，为使今后的工作取得更大的进步，现对本学期教学工作总结如下： 一、以课堂教学为核心： 1．备课。学期初,我钻研了《数学课程标准》,教材、教参,对学期教学内容做到心中有数。学期中,着重备好每一节课,掌握每一部分知识在单元中,在整册书中的地位,作用.思考学生怎样学,学生将会产生什么疑难,该怎样解决。多与段老师和闫老师沟通交流，向她们请教上课的方法，怎样讲授知识的方法能让孩子们接受。在备课本中体现教师的引导,学生的主动学习过程.充分理解课后习题的作用,设计好练习。 2．上课 （1）创设各种情境,激发学生思考。然后,放手让学生探究,动手,动口,动眼,动脑.针对教学重,难点,选择学生的探究结果,学生进行比较,交流,讨论,从中掌握知识,培养能力.接着,学生练习不同坡度,不同层次的题目,巩固知识,形成能力,发展思维.最后,尽量让学生自己小结学到的知识以及方法.现在学生普遍对数学课感兴趣,参与性高,为学好数学迈出了坚实的一步。 (2)及时复习。根据爱宾浩斯遗忘规律,新知识的遗忘随时间的延长而减慢.因此,我的做法是:新授知识基本是当天复习或第二天复习,以后再逐渐延长复习时间.这项措施非常适合低年级学生遗忘快,不会复习的特点。 (3)努力构建知识网络。一般做到一小节一整理,形成每节知识串;每单元整理复习形成知识链,一学期对整册书进行整理复习.学生经历了教材由"薄"变"厚",再
变量：属性地逻辑组合，通常用定义来解释或限定.自变量—实验条件（操控）; 调节变量：所要解释地是自变量在何种条件下会影响因变量，也就是说，当自变量与因变量间地相关大小或正负方向受到其它因素地影响时，这个其它因素就是该自变量与因变量之间地调节变量.中介变量：不可观察地，而在理论上又是影响所观察现象地因素 因变量—所需测定地特征或方面（可测量）额外变量—对因变量有一定影响，但与该次实验研究目地无关地变量（控制：屏蔽、中和））随机误差：可见误差.偶然地、随机地无关变量引起，较难控制，无规律可循；影响信效度.系统误差：常定误差.常定地、有规律地无关变量引起，其方向和大小地变化是恒定而有规律地.影响效度. 无关变量地控制：、消除法排除或隔离无关变量对实验效果地影响.标准化指导语、双盲程序、内隐测量、恒定法实验期间，尽量使得所有地实验条件、实验处理、实验者及被试都保持恒定.研究在同一时间、地点举行，程序、拉丁方设计、平衡法设置使得无关变量对所有地实验组和对照组地影响都均等、统计控制：一种事后补救，统计隔离无关变量地影响,协方差分析，偏相关.操作定义：描述所界定地变量或事项如何测量，包括：工具，方法，程序.将变量或指标地抽象称述转化为具体地操作称述地过程. 、简单随机取样（不作任何预处理，适用范围：对总体中各类比例不了解或来不及了解地情况）) 抽彩法（充分搅匀））随机表法（随
机进入））随机函数法（种子问题） 、分层随机取样：对总体进行预处理，分成若干层次后然后独立地从每一层次中选取样本. (）比例分层取样：按每一层次个体数量占总体中地比例决定该层次样本地数量. ()非比例分层取样：每层次中样本量不按该层次在总体中地比例抽取，而是根据研究者对不同层次个体地研究兴趣和侧重程度确定比例地大小. 、内部效度（逻辑度与额外因素影响度）：研究中自变量与因变量因果关系地明确程度. 影响因素：成熟因素、历史因素、被试选择上地差异、 研究被试缺失产生地效应、前测地影响、实验程序不一致或处理扩散产生地效应、统计回归效应、研究条件与因素间地交互作用.、外部效度：可细分为总体效度和生态效度.研究结果能一般化或普遍化到样本来自地总体（总体效度）和其它变量条件、时间和背景（生态效度）中去地程度，即研究结果地普遍代表性和适用性.影响因素：.取样代表性；.变量地操作方式不准，致使研究地可重复性差；.研究对被试地反作用；.事前测量与实验处理地相互影响；.多种处理地干扰；.实验者效应.研究地人为性；.被试选择与实验处理地交互作用.提高方法：严格控制；做好取样工作，包括被试取样、实验情境、研究工具、
何寨中心小学一年级数学期末试卷分析 学期已结束了，我以诚恳的工作态度完成了期末的数学检测工作。现将年级本期的数学检测卷面评析简析如下： 一、基本情况 本套数学试卷题型多样，容覆盖面广，题量恰当，对于本学期所学知识点均有安排，而且抓住了重点。本次期末考试共有39人参加,及格率92.11％,优秀率92.11％,全班最高分100分，平均分82.92分。 二、学生答题分析 1、学生答题的总体情况 对学生的成绩统计过程中，大部分学生基础知识扎实，学习效果较好，特别是在计算部分、图形的认识,这部分丢分较少。同时，从学生的答卷中也反映出了教学中存在的问题，如何让学生学会提出问题、分析问题、并解决问题，如何让我们的教育教学走上良性轨道，应当引起重视。从他们的差异性来分析，班级学生整体差距比较大的，说明同学之间还存在较大的差距，如何扎实做好培优辅差工作，如何加强班级管理，提高学习风气，在今后教育教学工作中应该引起足够的重视。本次检测结合试卷剖析，学生主要存在以下几个方面的普遍错误类型： 第一、不良习惯造成错误。学生在答题过程中，认为试题简单，而产生麻痹思想，结果造成抄写数字错误、加减号看错等。
第二、审题不认真造成错误。学生在答题过程中，审题存在较大的问题，有的题目需要学生在审题时必须注意力集中才能找出问题，但学生经常大意。 2、典型题情况分析 （1）填空题：学生对填数和数物体掌握较好，但在第4小题找规律填数、第7小题59.90元表示（）元（）角这几道题失分较多，学生在理解59.90元表示什么的这方面还有一定的困难。 （2）算一算：有20以的退位减法、两位数加减整十数、两位数加、减一位数（进位加、减），还有小括号的认识，这部分计算学生能够有效掌握计算方法，总体失分在2分左右，一小部分同学在这一块失分主要是马虎大意，看错+、-符号，另外还有个别同学在计算技能上稍有欠缺。 （3）比一比:主要是考查两位数比较大小，此外还对人民币的认识知识略有涉及，考查了人民币单位换算及大小比较，学生基本上都能够正确解答，这部分失分较少。 （4）选一选：在合适的答案下面打“√”，这一题考查学生对“多一些”“多得多”“少一些”“少得多”之间的理解，试卷上出现“接近”这个词语时，部分同学不能够理解这个词语的意思，导致失分，看来学生思维还不够灵活，平时还应做到举一反三。 （5）做一做：这部分有5道小题，考查学生的解决问题的能力。第1小题帮妈妈购物，学生失分较多的在④题，在理解题目意思上还
第二章设计心理学的研究方法 2.1 设计心理学的研究方法 人的消费活动是一种复杂的社会行为, 是人类心理活动的一部分。研究消费者心理活动规律的方法与整个心理学的一般研究方法是一致的, 心理学本身的发展, 为心理学应用分支的发展提供了科学的基础。但人类的消费活动是一种特殊领域。在运用心理学的某些研究方法了解消费行为规律时, 必然有一些新的内容和新的问题。因此探索设计心理学研究方法, 不仅有利于自身的发展, 也丰富了心理学主干研究方法的积累。设计心理学一般常用的研究方法有观察法、访谈法、问卷法、投射法、实验法、态度总加量表法、语义分析量表法、案例研究法、心理描述法、抽样调查法、创新思维法1 1 种方法。 2.1.1 观察法 观察法是心理学的基本方法之一。观察是科学研究的最一般的实践方法, 同时也是最简便、易行的研究方法。所谓观察法是在自然条件下, 有目的、有计划地直接观察研究对象( 消费者) 的言行表现, 从而分析其心理活动和行为规律的方法。设计心理学借助观察法, 用以研究广告、商标、包装、橱窗以及柜台设计等方面的效果。例如, 为了评估商店橱窗设计的效果, 可以在重新布置橱窗的前后, 观察行人注意橱窗或停下来观看橱窗的人数, 以及观看橱窗的人数在过路行人中所占的比例。通过重新布置前后观看橱窗的人数变化来说明橱窗设计的效果。 观察法的核心, 是按观察的目的, 确定观察的对象、方式和时机。观察时应随时记录消费者面对广告宣传、产品造型、包装设计以及柜台设计等方面所表现的行为举止, 包括语言的评价、目光注视度、面部表情、走路姿态, 等等。 观察记录的内容应包括: 观察的目的、对象, 观察时间, 被观察对象的有关言行、表情、动作等的数量与质量等。另外, 还有观察者对观察结果的综合评价。 观察法的优点是自然、真实、可靠、简便易行、花费低廉。在确定观察的时间和地点时, 要注意防止可能发生的取样误差。例如, 在了解商店消费者的构成时, 要区分休息日和非休息日, 也要区别上班时间和下班时间。有时商店消费者的构成也受周围居民成分的影响, 要观察少数民族消费者的特点, 就应该选择少数民族特需品的供应商店。在分析观察结果时, 要注意区分偶然的事件和有规律性的事实, 使结论具有科学性。 观察法的缺点也是明显的。在进行观察时, 观察者要被动地等待所要观察的事件出现。而且, 当事件出现时, 也只能观察到消费者是怎样从事活动的, 并不能得到消费者为什么会这样活动, 他的内心是怎样想的资料。 现代科技水平的发展, 使观察法能借用先进的观察设备诸如录像、录音、闭路电视的方式进行观察, 使观察效果更准确更及时, 并节省观察人员。但观察法只能记录消费者流露出来的言行、表情, 而对流露出这种言行、表情的原因, 是无法通过观察法直接获取, 因而必须结合其他的有关方法, 才能进一步了解消费行为规律。当研究的心理现象不能直接观察时, 可通过搜集有关资料, 间接了解消费者的心理活动, 这种研究方法叫调查法。调查法分为两种: 一种是口头调查法, 亦称谈话法、访谈法; 另一种是书面调查法, 亦称问卷法、调查表法。 2.1.2 访谈法 访谈法是通过访谈者与受访者之间的交谈, 了解受访者的动机、态度、个性和价值
2019—2020上学期期末小学一年级数学质量分析报告 一、试卷整体情况分析 本次数学考试试卷共五道大题。本次试卷充分体现了以教材为主的特点，能依纲据本，所考内容深入浅出地将教材中的所学内容展现在学生的试卷中，并注重考查学生活学活用的数学能力。适当关注学生综合运用知识分析和解决问题的能力。本套试卷难度不大，注重对基础知识基本技能的考查，并且题目灵活多样，评分详细合理。 二、试题考核目的分析 本次小学一年级数学试卷充分依据小学数学课程标准中的内容，紧扣教材又不拘泥于教材，整张试卷贴近学生生活，既考查了学生对数学基础知识的掌握，又检测了学生的运用能力。切合一年级学生的年龄特点，图文并茂，考查了学生第一学期所学知识的掌握情况。力求做到通过测试，了解各类学生真实学习状况，为教师改进教学提供较为可观的依据，为今后教学指明了方向。 三、试题考查结果及分析 （一）整体情况分析 本次参加考试的一共有340人，总分32415，人均分：95.34分，及格率为99.12%，优秀率为97.35%。由此得出，我镇一年级整体水平比较理想。全镇有6个全部及格的班级，全镇平均分最高的是方山一一班99.16分，最低的是方山一三班85.17分，优秀率最高的是100%，最低的是方山一三班66.67%。三率综合最好的班级是方山一一班10分，最低的是方山一三班7.96分，存在较大，只要努力，还
是有可进步的空间。由此可以看出，我镇一年级数学学科优秀率有待提高，学习数学的能力有待加强。一年级时起始阶段，我们的责任重大，应该进行认真反思。 （二）答题情况分析 第一题：算一算。学生出现个别错误，学生的口算能力一般，准确率不高。 第二题：填一填。主要考察学生对数的组成、数的顺序排列、比较大小、位置的掌握情况。题目紧扣课本、题型灵活。从整体看，大部分孩子做得都非常好，没出现错题，基础知识掌握比较好。错误率较高的是：（1）按规律填数：学生对20以内的单数认知掌握不够。（2）考试结束后，我们从元月9日到元月15日休息，我们打算休息（）天。此题学生不理解题意。 第三题：选择正确答案的序号填在括号里。出错的问题如下：（1）上下楼梯靠（）边行走，学生对左右分不清，导致失误。 第四题：动手操作。本题是对凑十法、钟面和物体的认识的考察。学生掌握比较好。 第五题：解决问题。能按解题思路去做，但答语不够完善。 （三）存在的问题 通过本次考试，感觉孩子存在以下几个问题： 1、书写不规范，拿起试卷看起来孩子写得挺干净的，但仔细观察会发现很多书写不到位。 2、学习习惯不太好，不认真倾听是孩子考不好的主要原因，本
1、准实验设计：由库克和坎贝尔提出。伴随现场实验的发展而发展起来。 →特点：介于真实验设计和非实验设计之间。从研究中对额外变量的控制程度出发，准实验设计差于真实验，但比非实验设计控制严格，一般准实验设计不容易随机挑选、分配被试，可能产生额外变量与自变量的混淆。另外，其外部效度不一定好。 →类型：①单组设计：时间序列设计、相等时间样本设计 ②多组设计：不相等组前后测设计、不相等组前后测时间序 列设计 2、实验研究： 定义：是在控制无关变量、操纵自变量的条件下研究心理现象的变化，从而确定变量之间关系的研究。对变量间的因果关系感兴趣。 特点：系统操纵或改变一个变量，观察这种操纵或改变对另一变量所造成的影响，并在此基础上揭示变量间的因果关系。 3、决定论：认为自然界和人类社会普遍存在客观规律和因果联系的理论。（不同于宗教的先决论） 4、被试变量：又称被试特性，一般作为自变量处理。 一是被试固有的。（如性别、婚姻状况、年龄、文化程度等） 二是暂时的。（如遭遇灾害否、吸毒否） 三是被试行为分类。（如喜欢早上锻炼还是喜欢晚上锻炼） 5、检查点：自变量的不同取值。 6、额外变量：会对结果产生影响，但是又不是研究者能够控制的变量。 7、混淆变量：无关变量的一种，该变量与自变量之间存在系统性关联，并对因变量产生影响的变量。这个影响的结果叫虚假效应。（PS.无关变量：与研究目的无关的变量） 8、因果关系研究：一个变量为操作变量，另一个为随自变量变化的因变量，可以揭示因果关系。 9、相关研究：两个变量都没有任何操作，只是调查、测量、观察的结果。如两个变量均为被试变量。（eg.与父母共处的时间与儿童的抑郁水平） 注：相关研究对个体差异感兴趣，而因果研究关心某类人的普遍心理规律；相关研究只研究机体间的方差，因果研究关心处理间的方差。 10、交互作用：反映的是两个或多个因素的联合效应。或者说是因素间相互影响和制约的关系。（当一个因素如何起作用受另一个因素的影响时，我们就认为存在二重交互作用。同理，当一个因素如何起作用受另两个因素的影响时，我们就认为存在三重交互作用。） 11、简单效应：指一个因素的水平在另一个因素某个水平的变异。（如，噪音大小与竞争对学习效果的影响，只考虑在竞争组噪音大小的简单效应） 12、简单简单效应：指一个因素的水平在另外两个因素的水平结合上的效应。（如，只考虑噪音强度在高难度和无竞争结合上的简单简单效应） 13、被试间设计：在不同被试之间比较。研究变量为被试变量时只能进行这种设计。 14、被试内设计：每个被试接受全部的水平或者水平结合的处理，自己和自己比较。 条件1 条件2
第一编心理学研究基础 第一章心理学与科学 1.心理学是一门兼有自然科学和社会科学的性质、具有不同分析单位以及既重视内省也重 视客观观察的学科。 2.一般人探索世界的常用方法归纳起来有常识、传统和权威三种。 3.系统性、重复性、证伪性和开放性是科学研究的四种典型特征。 4.心理学研究主要包括5个基本步骤：（1）选择课题和提出假设；（2）设计研究方案；（3） 搜集资料；（4）整理和分析资料；（5）解释结果以及检验假设。 5.科学研究的开放性主要表现在多视角、公开性、可争辩性以及科学研究无禁区四个方面。 6.以人为被试的研究的伦理规范主要包括知情同意、保护被试以及保障退出自由与保密。第二章选题与抽样 1.心理学研究课题可以有多种来源。主要的来源是：（1）对日常生活的观察；（2）实际的 需要；（3）理论；（4）研究文献的提示；（5）技术发展的推动等；也可以是这些来源的某种结合。 2.好的研究问题应该具备4个基本特征：（1）问题是切实可行的；（2）问题是清楚的；（3） 问题是有意义的；（4）问题是符合道德的。 3.文献是记录、保存、交流和传播知识的一切材料的总称。心理学文献就其保存方式可分 为纸质文献和互联网文献。纸质文献的常见类型有图书、期刊、会议和学术论文等。
4.查阅文献常用参考文献查找法、检索工具法。检索工具法分为手工检索和计算机检索。 5.取样和选题的关系密切，要依据课题的性质和研究结果的概括程度来选择被试。取样的 方法很多，根据是否按随机原则进行操作，取样可分为两类：非随机取样和随机取样。 6.简单随机取样又称纯随机取样，即对研究总体单位不进行任何分组排列，只按随机原则 直接从总体中抽取一定的样本，以使总体的每一个样本都有被同等抽取的可能性。分层随机取样，也称比率取样、分类取样和分组取样，是按照总体已有的某些特征，将总体分成几个不同的部分（每个部分叫一个层或一个子总体），然后在每一个层或子总体中进行简单随机取样。 7.系统取样，又称等距取样或机械取样，是以某种系统规则来选择样本的方法。聚类取样， 又称整群取样，是将总体按照某种标准（如班级、地区）划分为若干子群体，每个子群体作为一个取样单位，用随机的方法从总体中抽取子群体，将抽中的子群体中的所有单位合起来作为总体的样本。 8.多段取样，也称多级取样、分段取样，先将总体按照某种标准（如学校、地区）分为若 干层（组），称为第一层，然后从这一层中随机抽取出几个层作为第二层，依次类推，直至从最后一层中用随机取样的方式中抽取出一定数量的样本作为研究所需样本的取样方法。 9.方便取样：也称偶遇取样，研究者选择在方便的时间和地点将所遇到的人作为研究样本 的方法。立意取样，也称主观取样、判断取样，研究者根据自己的经验，如对总体构成要素和研究目标的认识，主观判断选取可以代表总体的个体作为样本。 10.定额取样，也称配额取样，是按特定的标准（性别、年龄、职业、受教育程度、教育 背景等）将总体中的个体分为若干类或层，然后在各层中取样。滚雪球取样，是先从总