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number of other problems can be effectively solved only based on their synthetic joint consideration. For example, for the translation of the word-combination “up-link communication” not into the English language as “communication with a satellite” it is necessary to use a visual image of the facts discussed in the text. This way, in a system of translations, while the text translating is a semantically visual image of narration that should be grown, a translation without a latter close to the human one is impossible.
For the realization of the principles aforementioned, it is necessary to start the elaboration of the dictionary for semantic trees of heterogeneous data: images, patterns of sentences composition, algorithms, and so on. It will be necessary to use the already existing dictionaries of sentences generation in the “Electronic Dictionary” software for the system basis.
In conclusion it is necessary to mention that generative grammar over the trees of strings is an effective means of generating the state trees for such systems, like natural language sentence and semantically loaded images. It is thought to apply the generative grammars over the trees of strings on the basis of the Semantic Trees’ Dictionary, which is a classification of heterogeneous semantic data.
References
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2. Apresyan Yu. D. Ideas and Methods of Modern Structural Linguistics. M. : Science, 1966.
3. Verdieva Z. N. Semantic Fields in the Modern English Language. M. : Higher School, 1986.
4. Lichargin D. V. Operations over the Natural Language Words Semes in Machine Translation // Works of the Conf. of Young Scientists. Krasnoyarsk : ICM SB RAS, 2003. P. 23-31.
5. Lichargin D. V. Elimination of Semantic Noise as the Means of Adequate Translation // Questions of the Theory and Practice of Translation. Works of All-Russian Conference. Penza : Privolzhye Region Knowledge House, 2003. P. 90-92.
6. Lichargin D. V. Generation of the Natural Language Phrases within the Task of Creating Natural Language Interface with Software // Problems of the Territory Information Development : Materials of the Eighth All-Russian Conf. PTID. 2003. Vol. 2. Krasnoyarsk : IP Centre of Krasnoyarsk State Technical University, 2003. P. 152-156.
7. Nikitin M. V. Lexical Meaning of a Word. M. : Higher School, 1983.
© Lichargin D. V., 2010
P. K. Lopatin
Siberian State Aerospace University named after academician M. F. Reshetnev, Russia, Krasnoyarsk
AN ALGORITHM FOR AN OBJECT GRASPING BY A MANIPULATOR IN AN UNKNOWN STATIC ENVIRONMENT
An algorithm for a n-link manipulating robot (MR) control in an environment with unknown static obstacles is considered. A theorem is proved which states that following the algorithm a MR in a finite number of steps will either grasp an object or will give a proved conclusion that an object cannot be grasped in any configuration.
Keywords: robot, unknown environment, obstacles, reachability.
In MR control the following typical problem arises: a MR should move from a start configuration q0 and grasp an object Obj by its gripper. Herewith sometimes the Obj may be grasped not in one but in several and sometimes in an infinite number of target configurations qj The target configurations are united into a target set BT. The set BT has an arbitrary shape.
Let us consider that the BT does not grow during the whole movement of MR. Consider also that the coordinates of every point from BT are known and defined reliably.
A MR is represented in the configuration space (generalized coordinate space) as a point. MR functioning should take place in the bounded region X of the configuration space. Let’s consider that X is such that for any q e X the following inequalities are fulfilled:
a1 < q < a2,
(1)
where a1 = (ai1, a2\ ..., a„) is a vector of
lower bounds on the generalized coordinates values, a2 = (a12, a22, . , an2) is a vector of upper bounds on the generalized coordinates values of a MR, q = (q1, q2,..., qn) is a vector of the generalized coordinates of a MR. So X is a hyper parallelepiped. We will consider all points not satisfying (1) as forbidden.
Moreover, it is necessary to take into account that there also may be forbidden states inside X. Firstly these are the states (configurations) conditioned by constructive limitations of a MR, for example, those in which inadmissible intersection of MR links takes place. It is possible to calculate such forbidden configurations in advance. Secondly we will consider a configuration as forbidden in case when it intersects obstacles. It is impossible to calculate all such configurations in advance in the conditions of an unknown environment. So we will consider a configuration as forbidden if a MR cannot be
present in it because of constructive limitations or because of intersection with an obstacle. Before the MR movement beginning we do not have information about forbidden states in X or it is incomplete. If we do not have exact information that a point q e X is forbidden, we will consider such point as allowed.
Now let us consider points from BT. We will consider a point qT e BT as allowed if it satisfies both criteria: 1) it is not forbidden in senses described in previous paragraphs, 2) it may be reached from q0 in a finite number of steps moving in X by allowed states. We will consider points from BT as forbidden if they do not satisfy at least one criterion. As far as we need to find out whether the set BT is reachable at least in one point we will consider that before the MR movement we do not have information about any point from BT whether it is forbidden or allowed. Now let us formulate the following Problem of a MR control in an unknown static environment: a start configuration q0 and a target set BT are given. It is necessary to propose an algorithm which in a finite number of steps will either move the MR from q0 to a point from BT or will give a proved conclusion that there is no allowed state in BT.
Review of related works. Currently there are many works dedicated to algorithms for dynamic systems (DS) control and, in particular, for robotic systems (RS) in known and unknown environments. There are good reviews of such algorithms [1; 2]. There are algorithms, for example, the forward search algorithm and the A* algorithm [3] which in a finite number of steps will either find a path from q0 to a point from BT or inform that there is no point in BT reachable from q0.
Some algorithms for planning in a known environment in principle may be used for movement in an unknown environment. If we discretize the configuration space then we may use graph algorithms for a DS path searching [2; 3]. But these algorithms have one common feature which makes their application for the DS control in an unknown environment very difficult. The feature is that they demand to carry out the breadth-first search in a certain volume otherwise reaching of a target point qiT is not guaranteed [4]. But during the breadth-first search the following situation often arises: suppose we have just finished considering the vertices adjacent to a vertex q and we have to consider vertices adjacent to a vertex q ’ and the q and q ’ are not adjacent. In order to consider vertices adjacent to the q’ the manipulator has to come to the q ’ at first. So we get a problem of the manipulator movement from q to q ’. The necessity of searching and following paths for multiple different q and q’ makes the total sum of the manipulator movements very big [4]. In case when we plan a path in a known environment a computer simply “switches its attention” from q to q ’, which are stored in its memory.
According to classification [2] it is possible to outline the following representatives of the breadth-first approach: proper breadth-first searching algorithm, A* algorithm, best-first heuristic search, lazy PRM, dynamic
programming. The methods based on a randomized potential field, Ariadne’s Clew algorithm, rapidly-exploring random trees [2] have such feature that new vertices are generated randomly and therefore using these methods for the unknown environment leads to the same difficulties. The approaches based on cell decomposition, visibility (bitangent) graphs, Voronoi diagrams [2] are reduced to alternate graph building and searching a path on it and have the above mentioned disadvantage connected with multiple mechanical movements.
In the algorithm presented in this article the vertices q and q’ are always neighbor vertices and it reduces the number of movements.
It is also known that the “depth-first” algorithms do not guarantee reaching the goal [4].
There is a common difficulty for the methods of path planning in the presence of known obstacles: it is very difficult to borrow full information about workspace of a robot in advance and to represent this information in a form suitable for trajectory planning. Considering our algorithm one can see that there is no need for a control system to have full information about workspace in advance, a manipulator will borrow necessary information by itself in limited quantities and in terms of generalized coordinates which is suitable for path planning.
The attempts of creating algorithms for the robot control in presence of unknown obstacles were made. Most of them cover various two-dimensional cases [5].
In [6-9] different approaches for a robot control in two-dimensional unknown environment are considered. In [6; 9] the approaches are based on Voronoi diagrams, in [8] a tabu search approach is presented. As these approaches demand alternate graph building and searching a path on it they lead to multiple robot movements. In [7] obstacles should have polygonal form. The application of methods proposed in [6-9] to a n-link manipulator control in an unknown environment is not presented.
In [5] an algorithm for the control of manipulators in the presence of unknown obstacles in three-dimensional space is given. A MR must have not more than three elements and the last kinematic pair should be sliding. In the given conditions the algorithm in a finite number of steps either transfers a MR into a target configuration or informs that it is unreachable.
In [10] the n-dimensional case is considered. The algorithm is based on the solution of the system of nonlinear equations using Newton method and therefore it cannot guarantee the reaching of a target position.
In [2] algorithms for moving a robot in the presence of uncertainty (including cases of an unknown environment) are considered. The algorithms are based on the sequential solution theory. In general the algorithms do not guarantee reaching the goal. In cases when the algorithms use searching on a graph the above mentioned difficulty arises connected with multiple mechanical movements.
In [11] an algorithm for controlling dynamic systems in an n-dimensional state space in presence of unknown forbidden static states is proposed. The algorithm in a finite number of steps either moves the DS from a start point q0 to a target point qT or gives a proved conclusion that the qT is unreachable. The algorithm’s disadvantage is that it assumes that in the set X any state may be reached from any state. In a number of cases such demand may contradict the purpose of the algorithm to define in a finite number of steps whether a target state is reachable from a start state.
In [4] an approach to robot’s (including manipulating robots) control in a n-dimensional state space is proposed. The essence of this approach is that the robot generates a path, connecting a start point q0 and a target point qT, not intersecting known forbidden states and tries to follow the trajectory either till reaching the qT or till meeting an earlier unknown forbidden state in a point qn. In the last case a new path L(qn, qT) is generated, connecting qn and qT and not intersecting any known forbidden state. It is shown [4] that the problem of the robot movement from q0 to qT in an unknown environment is reduced to a solution of a finite number of problems of generating and following the path L(qn, qT) (in other words to a solution of a finite number of problems PI (planning in a known environment)). Herewith it is supposed that we have a priori information that the qT is reachable.
In this article we used the approach [4] for a solution of the Problem. The Problem is reduced to investigation of reachability of a finite number NBT of points qiT, I = 1, 2, ..., NBT. The demand of the a priori knowledge of qiT, I = 1, 2, ., NBT reachability is omitted. New demands to the PI procedure necessary for the Problem solution were formulated.
In [12-15] algorithms were given which in a finite number of steps move a DS from q0 to qT in an environment with unknown static states. Herewith it was supposed that there is a priori information that qT is reachable.
Preliminary conditions
1. Let us extract from the set BT a finite number NBT of points qiT, I = 1, 2, ..., NBT. These are the configurations whose reachability will be explored. Further we will consider BT as a list of configurations qiT, I = 1, 2, ., NBT Let’s consider that BT is not replenished and therefore NBT is not increased, consider that coordinates of every point from BT are defined reliably.
2. Consider that we have a procedure PI which in a finite number of steps either generates a path from an arbitrary allowed point qne X to an arbitrary point qTe X in the presence of known forbidden states or informs that qT is unreachable. Such procedures already exist, for example, the forward search algorithm or the A * algorithm [3], which for any start point qn and any target point qT under given known forbidden states in a finite number of states either find a path from qn to qT or inform that a path from qn to qT cannot be found.
3. The obstacles’ disposition inside the MR working area does not change during the whole time of the MR movement.
4. The obstacles’ number inside the MR working area does not change during the whole time of the MR movement.
5. The MR movement including the resultant path should take place inside the hyperparallelepiped (1).
6. The MR has a sensor system (SS) which supplies information about a r-neighborhood of a current MR point qne X. The current point of the MR is the point where the MR is situated right now. The r-neighborhood of the q is a hyperball in X with a center in q and radius r > 0. We denote the set of all points comprising the r-neighborhood of the q as Y(q). The words “supplies information about the r-neighborhood of the point q“ mean that the SS defines whether every point from Y(q) is allowed or forbidden. Herewith all forbidden points from Y(q) are stored in a set Q(q) and all allowed points from Y(q) are stored in a set Z(q). There may be different ways of the sets Y(q), Q(q), Z(q) representation - in a form of formulas, lists, tables etc., but we consider that we have such representation. We will not consider the SS structure.
7. Consider that we have a program Procedure1(BT, NBT, Q(qn)). Procedurel(-) in the moment of call gets the set BT, the number NBT of points in the set BT, the set Q(qn) which was formed during the last call of the SS. Procedurel(-) throws out of the BT those points which co incide with points from Q(qn). After the throw the points left in the BT are renumerated by the continuous numeration beginning from 1 and the number of points left in BT after execution of Procedurel(-) is inscribed in
NBT .
8. Consider that we have a program Procedure2(BT, NBT, qT). Procedure2(-) in the moment of call gets the set BT, the number NBT of points in the set BT and the point qT. Procedure2() throws out of the BT the point qT. After that the points are renumerated by the continuous numeration beginning from 1 and the number of points left in BT after execution of Procedure2() is inscribed in NBT.
Below the Algorithm for the solution of the Problem is given. Before a movement beginning the current configuration qc of the MR is q0, during the movement the Algorithm 1 may be called from other current configurations of the MR.
Algorithm
If NBT = 0 then the Algorithm terminates its work with a message that the Obj cannot be grasped. Otherwise we consider that the first point from BT is qT.
STEP 1. The MR is in a configuration qc. n: = 0, qn: = qc. Execute target_point_is_forbidden: = Algorithm 1 (qc, qT, BT, NBT). If target_point_is_forbidden: = NO then the Algorithm successfully terminates its work with a message that the object is grasped in the point qT.
If target_point_is_forbidden = YES go to STEP 2.
STEP 2. If NBT = 0 the Algorithm terminates its work with a message that an object cannot be grasped in any target configuration. If NBT ^ 0 consider the first point from BT as qT and go to STEP 1. The End of the Algorithm.
Algorithm 1 gets values in the format: Algorithm 1(qn, qT, BT, NBT) and defines whether point qT is reachable from qn in an unknown environment or not.
Algorithm 1
STEP 1. MR is in qn (let us call it “a path changing point”). SS supplies information about Y(qn), Z(qn), Q(qn).
STEP 2. Execute
NBT:= Procedure1(BT, NBT, Q(qn)).
If NBT = 0 then target_point_is_forbidden: = YES and return to the Algorithm. If NBT 4 0 check whether the point qT was thrown out of BT. If yes then target_point_is_forbidden: = YES and return to the Algorithm, if no (that is qT was left in BT) go to STEP 3.
STEP 3. Call the PI procedure in order to generate a path L(qn,qT) satisfying the following conditions:
- L(qn,qT) connects qn and qT;
n
- L(qn,qT) does not intersect the set U Q(qn) that is, it
i =0
does not intersect any forbidden point;
- L(qn,qT) satisfies the limitations (1).
There may be two results of the PI procedure execution:
1. PI returns the generated path and Algorithm 1 goes to STEP 4;
2. PI informs that the L(qn,qT) cannot be generated, that is, the qT is unreachable. In this case:
NBT:= Procedure2(BT, NBT, qT),
make assignment target_configuration_is_forbidden: = =YES and return to the Algorithm.
STEP 4. MR begins to follow the L(qn,qT). There may be two results:
1. MR comes to a point qJeBT. In this case make assignments qT: = qj and target_point_is_forbidden: = NO and return to the Algorithm;
2. MR will come to such point q that the next point after that is forbidden. In this case execute n: = n + 1, qn: = q* and Algorithm1 goes to STEP 1. The End of Algorithm 1.
Theorem. Executing the Algorithm the MR will solve the Problem in a finite number of steps.
Proof. The Algorithm defines the reachability of a finite number of points from BT. In order to define whether a point qiTe BT is reachable it is necessary to call Algorithm 1 one time. Therefore the Algorithm execution is reduced to a finite number of calls of Algorithm 1. Therefore in order to demonstrate that the Algorithm will be executed in a finite number of steps it is necessary to show that Algorithm 1 will be executed in a finite number of steps for arbitrary qn and qT.
Algorithm 1 defines whether a point qT is reachable in an unknown environment from a path changing point qn or not. In Algorithm 1 when the MR is in point qn, n = 0,
1, 2,. the SS and the PI procedure are called. If after executing of these actions the point qT will be defined as forbidden (because of intersection with obstacles or unreachability) the return to the Algorithm will take place and another point qiTe BT will be considered. If after execution of these actions the point qT will not be defined as forbidden, a path L(qn, qT) will be generated and the MR will begin to follow this path. There may be two results of following this path: either MR will not meet forbidden points and therefore will reach the qT (and therefore successful termination of the Algorithm work will occur) or the MR will come to such a point qn, n = 1,
2, . that the next point will be forbidden. Let us show that all path changing points qn, n = 0, 1, 2, . will be different and their number will be finite.
Let us prove that all points where the MR changes its trajectory will be different. Suppose that the manipulator changed its trajectory being in point qs, and later it again changed its trajectory, being in point qp, that is s<p. Let us show that qs ^ qp. Suppose, at first, that, on the contrary qs = qp . Then Q(qs) = Q(qp). As the manipulator changed its trajectory being in point qs it generated the trajectory which did not intersect the sets Q(qi), I = 0, 1, ... , s. But as it changed the trajectory in point qp it means that its trajectory intersected Q(q p) = Q(qs) (besides, qs = qp is the center of r-neighborhood of the point qs = qp and the following point is forbidden). That is, Q(q p) = Q(qs) was unknown. Here is a contradiction. It means that all points where the manipulator changes its trajectory are different.
Now let us show that the number of such points is finite. Suppose that it is infinite. All points of a trajectory changing must satisfy the inequalities (1). It means, that the sequence of these points is limited. According to the Boltsano-Weierstrass theorem it is possible to extract a convergent subsequence q, I = 1, 2, ... from this sequence According to Cauchy property of the convergent sequences it is possible for any e to find such a number s that all points qi, i > s will lie in an e-neighborhood of qs. Let us take e < r. Consider an arbitrary point qi of the trajectory changing lying in the e-neighborhood of qs. As the manipulator had to change the trajectory in the qi, it means that that trajectory intersected Q(qs) (because qi and its neighbor points belong to Q(qs)). From this fact it is possible to draw the conclusion that the set Q(qs) was not taken into account when that trajectory was generated. But such situation is impossible if we strictly follow the conditions of the Algorithm 1. The situation when a trajectory changing point belongs to the e-neighborhood of another trajectory changing point will necessarily appear if the number of the points where trajectory is changing is infinite. But we showed that such situation is impossible and it means that a number of the points where it is necessary to change trajectory will be finite.
So, the number of path changing points qn, n = 0, 1, 2,... is finite and they are different. In every point qn the SS and the procedure PI are called and PI generates a L(qn, qT). As a result we either get information that qT is forbidden or do not get it. If we get the information that qT is forbidden then we consider the qT as unreachable. Otherwise an attempt of the path L(qn, qT) following occurs. If in the last path changing point qn the point qT was not qualified as forbidden, a path L(qn, qT) will be generated, this path will be followed and the qT will be reached.
So it was shown that executing Algorithm 1 the MR will either reach the qT or will make the conclusion that the qT is unreachable. The Algorithm is reduced to a finite number of the Algorithm 1 calls. Therefore one may see that executing the Algorithm the MR will solve the Problem in a finite number of steps. The Theorem is proved.
Note 1. We have already mentioned that Algorithm 1 is reduced to a finite number of paths L(qn, qT) generation and following. The Algorithm is reduced to a finite number of the Algorithm 1 calls. Therefore one may see that the Problem is reduced to the solution of a finite number of PI problems of a path generating and following in an environment with known forbidden states.
Note 2. On the first call of the Algorithm 1 qc = q0, on the next calls, generally speaking, qc # q0. Upon the Algorithm 1 execution a conclusion about a qT reachability/unreachability from a qc is drawn. But, as the MR has come to qc from q0 by continuously following each other allowed points, the conclusion about reachability/unreachability of qT from qc will be simultaneously considered as the conclusion about reachability/unreachability of qT from q0.
An algorithm for a n-link manipulating robot control in an environment with unknown static obstacles was considered. A theorem was proved which states that following the algorithm the MR will either grasp an object or will give a proved conclusion that an object cannot be grasped in any configuration in a finite number of steps .
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© Lopatin P. K., 2010
