Robot motion similarity analysis using an FNN learning mechanism
Kuu-young Young
∗, Jyh-Kao Wang
Department of Electrical and Control Engineering, National Chiao-Tung University, Hsinchu, 30050, Taiwan Received 12 August 1998; received in revised form 21 March 2000; accepted 26 April 2000
Abstract
Learning controllers are usually subordinate to conventional controllers in governing multiple-joint robot motion, in spite of their ability to generalize, because learning space complexity and motion variety require them to consume excessive amount of memory when they are employed as major roles in motion governing. We propose using a fuzzy neural network (FNN) to learn and analyze robot motions so that they can be classi7ed according to similarity. After classi7cation, the learning controller can then be designed to govern robot motions according to their similarities without consuming excessive memory resources. c 2001 Elsevier Science B.V. All rights reserved.
Keywords: Robot learning control; Learning space complexity; Motion similarity analysis; Fuzzy neural network
1. Introduction
Learning controllers are able to tackle highly com-plex dynamics without explicit model dependence and identi7cation [7,13,20,21]. In addition, they are also considered capable of generalization [1,11]. However, learning controllers are usually used as subordinates to conventional controllers in governing robot motions [10,12,14]. The conventional controller is responsible for the major portion of the control, and brings the system close to the desired state, after which the learn-ing controller compensates for the remainlearn-ing error. Some learning control schemes do however use learn-ing controllers alone to execute motion control. But,
This work was supported in part by the National Science
Council, Taiwan, under grant NSC 87-2213-E-009-145. ∗Corresponding author. Tel.: 5712-121; fax: +886-3-5715-998.
E-mail address: kyoung@cc.nctu.edu.tw (K. Young).
most of them need to repeat the learning process each time a new trajectory is encountered [8]. This learning controller de7ciency results mainly from the complex-ity of motions associated with various task require-ments, e.g., diEerent movement distances, velocities, and loads. Consequently, when a learning controller is given a major role in governing the general motion of a multiple-joint robot manipulator, the learning space it must deal with is extremely complicated [15,17].
In order to simplify the complexity of the learn-ing space in uslearn-ing learnlearn-ing control to govern robot motions, we propose, in this paper, performing sim-ilarity analysis of robot motions using a fuzzy neu-ral network (FNN) learning mechanism to classify robot motions according to their similarities. The FNN is basically a fuzzy system that uses a neural net-work structure, such that the fuzzy system parameters can be adjusted automatically [3,5,11]. During anal-ysis, the FNN is 7rst used to learn to govern var-ious robot motions. The FNN may require diEerent
0165-0114/01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved.
numbers of rules and shapes of membership functions to govern each speci7c motion. The similarities be-tween motions are evaluated according to the numbers of rules and the shapes of the membership functions the FNN requires to govern. After classi7cation, robot motions can then be governed by using learning con-trollers which are allocated memory resources accord-ing to motion similarities. In particular, groups of robot motions with high degrees of similarity will de-mand learning controllers with smaller memory sizes because these motions correspond to similar fuzzy pa-rameters in the FNN. By contrast, when robot motions are randomly arranged, learning controllers with larger memory sizes will be needed to govern motions.
Application of the proposed similarity analysis on robot motion classi7cation can therefore lead to an organized and simpli7ed learning space for motion governing. In addition, because both the similarity analysis and motion governing use the learning mech-anism, motion analysis and governing are consistent, making the classi7cation more eEective. At the cur-rent stage of the study, the proposed approach is not ready for similarity analysis on general motions of general industrial robot manipulators. Motion features that can properly represent robot motion character-istics in learning for serving as motion classi7cation indices also remain to be found. Instead, the main focus of this paper is to demonstrate how to classify robot motions via the means of learning. The proposed motion similarity analysis is discussed in Section 2. Motion classi7cation based on similarity analysis is presented in Section 3. In Section 4, simulations based on use of a two-joint robot manipulator are reported. Finally, discussions and conclusions are given in Section 5.
2. Proposed motion similarity analysis
Motion similarity can be de7ned according to dif-ferent characteristics. For example, a number of arbi-trary robot motions can be categorized into classes of motions with similar movement distances, velocities, or loads [22]. However, this classi7cation cannot guar-antee that motions in the same class will correspond to similar fuzzy parameters when governed using an FNN. In the proposed approach, we aim to group sim-ilar motions to simplify the complexity in the learning
space. Therefore, from the standpoint of learning, in this paper, similar motion is de7ned as
Denition 1 (Similar motion). Two motions gov-erned using an FNN are said to be similar if the numbers of fuzzy rules they require are the same, and the similarity among the shapes of their correspond-ing membership functions is above a pre-speci7ed threshold.
According to De7nition 1, Fig. 1 shows the con-ceptual organization of the proposed motion similarity analysis. An FNN is used to learn to govern the entire trajectory of an input motion. Initially, a large num-ber of FNN linguistic labels are used in the learning. The learning process will terminate when the FNN can successfully govern the motion up to a pre-speci7ed accuracy. During learning, redundant fuzzy rules in the FNN are eliminated, and the 7nal FNN fuzzy rule number required and the corresponding membership function distribution for governing the motion are then determined. According to the FNN fuzzy rule numbers and the similarity between the membership functions by comparing the areas covered by the corresponding fuzzy sets, the degrees of similarity between motions are then obtained. Finally, the motions input in arbi-trary fashion are classi7ed into groups of motions ac-cording to their similarities.
For motion governing using an FNN in Fig. 1, the system in the block is not only with an FNN, but also includes a local controller connected in series with the FNN, as shown in Fig. 2 [22]. With this hierarchi-cal structure, the complexity in motion governing is shared by the FNN and the local controller at diEer-ent levels. By contrast, in previous approaches learn-ing controllers are usually connected in parallel with conventional controllers, and used as subordinates to conventional controllers [10,12,14]. It can be seen that without the local controller in Fig. 2, the FNN would have to govern the robot manipulator directly. In other words, the control signal from the FNN is in the torque level, and thus very sensitive to Iuctuations. There-fore, the system structure in Fig. 2 allows the FNN to function at a higher level, and thus generate more ab-stract, robust control signals [6,22]. Then, the fuzzy parameters in the FNN will be more signi7cant for similarity evaluation. Fig. 2 shows the reference po-sition and velocity trajectories, r and ˙r, of an input
Fig. 1. Conceptual organization of the proposed motion similarity analysis.
motion being fed into the FNN, which in turn gen-erates motion commands Cm and sends them to the
local controller. The local controller then modulates the motion commands via position and velocity feed-backs, and ˙, and uses the resultant torque to move the robot manipulator. In our design, only the desired position is speci7ed in the motion command. A sim-ple position control law with linear damping is then used for the local controller [19]:
= Kp(Cm− ) − Kd˙; (1)
where Kpand Kd are symmetric positive de7nite
ma-trices for stability considerations [16].
The system shown in Fig. 2 is used to derive the FNN fuzzy parameters for motion similarity evalua-tion. Those motions evaluated to be with high degrees of similarity can then be governed by using very sim-ilar fuzzy parameters. Those with medium degrees of similarity can have their fuzzy parameters general-ized to deal with a wider range of motions using a learning mechanism with a memory allocated accord-ing to the degrees of nonlinearity exhibited. In [22], we reported that a CMAC-type neural network can be used to generalize fuzzy parameters from sets of FNN fuzzy parameters appropriate for governing a number of sampled motions in a class to govern the whole class of motions. In some sense, the FNN fuzzy pa-rameter generalization implies that qualitative fuzzy rules are generalized, and it tends to cover a larger learning space. And those with low degrees of simi-larity may demand learning mechanisms with larger memory sizes for generalization. Learning controllers can thus be designed and allocated appropriate mem-ory sizes to govern the classi7ed robot motions with diEering degrees of similarity.
3. Motion classication based on similarity
Fig. 3 shows the block diagram of the proposed motion classi7cation scheme in which the FNN
learn-Fig. 2. Block diagram of motion governing using an FNN.
ing mechanism discussed in Section 3.1 is used in two learning processes involving motion classi7ca-tion. The 7rst learning process is intended to elimi-nate redundant linguistic labels for each input motion, which will make the resulting FNN structure more concise and allow evaluation of membership function distributions to be more meaningful. A second learn-ing process uslearn-ing an FNN with a new structure to ob-tain new membership function distributions that are then used for motion similarity measurement between input motions. Both learning processes require simi-larity analysis, as discussed in Sections 3.2 and 3.3, respectively.
The operation of the proposed motion classi7ca-tion scheme is as follows. As Fig. 3 shows, in the 7rst learning process a large number of FNN lin-guistic labels are initially chosen in arbitrary fash-ion and normal fuzzy sets are used as membership functions. The learning process terminates when the FNN can govern motion successfully; i.e., the posi-tion mean-square error (M.S.E.) is less than a pre-speci7ed value (e1). After the input motion has been
learned, the similarities between membership func-tions corresponding to this motion are evaluated pair by pair. When membership functions are very simi-lar, it indicates that some of the linguistic labels are unnecessary and can be eliminated. Therefore, after the 7rst learning process, the FNN will have a sim-pli7ed structure and be sent to the second learning process. The second learning process also terminates when the FNN can govern motion successfully; i.e., when M.S.E. ¡e2. The resulting membership function
Fig. 3. Block diagram of the proposed motion classi7cation scheme.
motion similarity measurement. Before the similarity measurement, membership functions are 7rst normal-ized to place the similarity evaluation of membership function distributions on the same scale. This is neces-sary because various input motions may correspond to diEerent ranges of movement distances and velocities. Since the normalization involves only linear ampli7-cation or compression of the membership functions, their characteristics are not altered. Accordingly,
in-put motions are classi7ed into groups of motions with diEering degrees of similarity.
3.1. The FNN learning mechanism
The FNN learning mechanism used in the proposed scheme is as shown in Fig. 4. The representation of a fuzzy system using a fuzzy neural network enables us to take advantage of the learning capability of the
Fig. 4. The structure of the FNN.
neural network for automatic tuning of the parameters in the fuzzy system. The fuzzy reasoning parameters are thus expressed in the connection weights or node functions of the neural network [3,5,11]. In the pro-posed scheme, we choose an FNN with a structure similar to that in [5], of course, other types of FNN can also be used. As Fig. 4 shows, the inputs to the FNN are position and velocity trajectories of input motions, r and ˙r, and the outputs are motion commands Cm.
There are four layers in the FNN: the input layer, the linguistic layer, the rule layer, and the output layer. Gaussian functions with adjustable means and vari-ances are used as membership functions. Parameters to be learned in this FNN are premise parameters in the second layer, representing the means and variances of the Gaussian functions, and consequence parame-ters in the fourth layer, representing the weights for
the consequence links connected to the output node. A gradient-descent-based back-propagation algorithm is employed for learning [13]. More detailed discussions of the structure and learning process of this FNN can be found in Appendix A.
3.2. Similarity measure for redundant rule elimination
To eliminate redundant rules in using an FNN to govern a motion, the similarities between member-ship functions after learning are evaluated pair by pair. The method for similarity measure of fuzzy sets proposed in [4,11] was adopted for evaluation. In [4,11], similarities between fuzzy sets are computed by comparing the areas covered by fuzzy sets according to geometric points. The similarity measure between
Fig. 5. Similarity measure of two fuzzy sets, A and B: (a) B ⊆ A, (b) A and B have one intersection point, (c) A and B have two intersection points, (d) A ∩ B = ∅.
two fuzzy sets A and B is thus de7ned as
E(A; B) = M(A ∩ B)M(A ∪ B); (2) where E(A; B) is the degree of similarity between A and B, ∩ and ∪ denote the intersection and union operators, respectively, and M(·) is the size of a fuzzy set, i.e., the area it covers. Note that 06E(A; B)61; E(A; B) = 1, when A = B, and E(A; B) = 0, when they do not intersect.
Because Gaussian functions are used as member-ship functions, the computation of the intersection and union in Eq. (2) involves two Gaussian functions with
nonlinear shapes. To simplify computation, the trian-gular function described in Eq. (3) is used to approx-imate the Gaussian function:
exp −(x − m)2 2 → 0; √ − |x − m| √ ; (3) where m and are the mean and variance of the Gaussian function, and m and √ the center and width of the triangular function, respectively. Via the approximation, the similarity measure between A and B can be divided into four cases, as shown in Fig. 5. Detailed discussions can be found in Appendix B.
3.3. Motion similarity measure
The similarity measure described in Section 3.2 is used to eliminate redundant rules for each input mo-tion, and is based on the area covered by two fuzzy sets. To evaluate similarities between motions, the measure is extended to deal with group of fuzzy sets corresponding to various motions. Thus, the areas cov-ered by all membership functions that govern input motions are used. In addition, the fuzzy sets need to be normalized to place the similarity evaluation of membership function distributions on the same scale, because various input motions may correspond to dif-ferent ranges of movement distances and velocities.
Assume there are two motions, m1and m2, having
several inputs. For a speci7c input i, the similarity measure between m1and m2 for input i is de7ned as
Ei(m1; m2) =MMi(m1∩ m2)
i(m1∪ m2); (4)
where Mi(m1∩ m2) and Mi(m1∪ m2) stand for the
intersection and union areas of the membership func-tions of m1 and m2 for input i. Fig. 6 shows an
ex-ample of two sets of membership functions, mm1and
mm2, having three linguistic labels, to which two
mo-tions corresponding to one speci7c input belong; the shaded areas are the intersection area between mm1
and mm2. By considering all the inputs and performing
a minimum operation (min) upon similarity measures de7ned in Eq. (4), the similarity measure between m1
and m2can be de7ned as
E(m1; m2) = mini {Ei(m1; m2)}: (5)
In Eq. (5), the use of the min operation guarantees that the similarity measures for all the inputs will be above certain threshold. As an example, the similar-ity measure between m1and m2 for a two-joint robot
manipulator with four inputs, r1, r2, ˙r1, and ˙r2, can
be found using
E(m1; m2) = min{ Er1(m1; m2); E˙r1(m1; m2);
Er2(m1; m2); E˙r2(m1; m2)}: (6)
4. Simulation
Simulations were performed to demonstrate the ef-fectiveness of the proposed motion similarity analysis
Fig. 6. Similarity measure of two groups of fuzzy sets with three linguistic labels.
Fig. 7. A two-joint robot manipulator.
based on use of the two-joint robot manipulator shown in Fig. 7. The dynamic equations for this two-joint robot manipulator are expressed as follows:
1= H11M1+ H12M2− H ˙22− 2H ˙1˙2+ G1; (7) 2= H21M1+ H22M2+ H ˙21+ G2; (8) where H11= m1l2c1+ I1 + m∗ 2[l21+ l∗2c2 + 2l1l∗c2cos(2)] + I2∗; (9) H22= m∗2l∗2c2+ I2∗; (10) H12= m∗2l1l∗c2 cos(2) + m∗2l∗2c2 + I2∗; (11) H21= H12; (12)
H = m∗ 2l1l∗c2 sin(2); (13) G1= m1lc1g cos(1) + m∗ 2g[l∗c2 cos(1+ 2) + l1 cos(1)]; (14) G2 = m∗2l∗c2g cos(1+ 2) (15) and m∗ 2 = m2+ M; (16) l∗ c2= m2lc2m+ Ml∗ 2 2 ; (17) I∗ 2 = I2+ m2(l∗c2− lc2)2+ M(l2− l∗c2)2 (18) with m1= 2:815 kg; m2= 1:640 kg; l1= 0:30 m; l2= 0:32 m; lc1= 0:15 m; lc2= 0:16 m; I1= I2= 0:0234
kg m2, and M represents the mass of the load. The eEect of gravity was ignored in the simulation. To provide various input motions, a second-order system was used, as described below:
L M + B ˙ + K( − d) = 0; (19)
where L is the load, K the stiEness, B the damp-ing coeNcient, and and d the actual and desired
joint positions for each joint, respectively. DiEer-ent motions were generated by varying L, B, K, d, damping ratio , and undamped natural
fre-quency Wn. For motion control, each joint of the
robot manipulator was equipped with an FNN and a local controller. The gains of the local controller in Eq. (1) were set to Kp= 30 and 8 N m=rad and
Kd= 5 and 1 N m=(rad=s) for joints one and two,
respectively.
In the 7rst simulation, applying the proposed mo-tion classi7camo-tion scheme shown in Fig. 3 resulted in two sets of motions being classi7ed as having high and low degrees of similarity, as shown in Fig. 8. Initially, the number of linguistic labels for all motions input to the FNN was set at 7ve, and thus there were 25 fuzzy rules in the rule layer of each FNN. In the 7rst learning process for eliminating redundant linguistic labels, the similarity threshold between membership functions was set at E ¿ 0:9. After redundant rule elimination was performed on all motions in Figs. 8(a) and (b), the numbers of linguistic labels for inputs, r1, r2, ˙r1, and ˙r2, reduced to 5; 5; 3; and 2,
respec-tively, for a total of 15 and 10 fuzzy rules for joints one
Fig. 8. (a) Motions with high degrees of similarity. (b) Motions with low degrees of similarity.
and two, respectively. Performing the second learn-ing operation on these motions resulted in the degrees of similarity listed in Tables 1 and 2, respectively. Table 1 shows the similarities among motions shown in Fig. 8(a) above 0.9; Table 2 shows those shown in Fig. 8(b) below 0.7. Note that if the numbers of linguistic labels for some motion inputs diEered from those of other motions, the motions were taken to be dissimilar directly, and did not need the second learn-ing phase.
In the second simulation, we intended to show that learning controllers with smaller memory allocations can be designed to govern motions with high degrees of similarity. Table 3 shows the corresponding means and variances (the premise parameters in the second
Table 1
Similarities between motions with high degrees of similarity in Fig. 8(a)
Motion number Motion number
1 2 3 4 5 1.0 0.963 0.966 0.943 0.921 1 1.0 0.927 0.938 0.962 2 1.0 0.975 0.953 3 1.0 0.917 4 1.0 5 Table 2
Similarities between motions with low degrees of similarity in Fig. 8(b)
Motion number Motion number
1 2 3 4 5 1.0 0.324 0.245 0.420 0.113 1 1.0 0.442 0.376 0.531 2 1.0 0.482 0.691 3 1.0 0.287 4 1.0 5
layer) in the FNN governing those motions with high degrees of similarity in Fig. 8(a). In Table 3, the dif-ferences between these means and variances are very small, indicating that the input membership function distributions are very similar. We thus designed a learning controller to govern the entire group of mo-tions in Fig. 8(a), in which the same set of premise parameters, i.e., the average means and variances cor-responding to the motions, were used for all the gov-erning FNNs and the consequence parameters in the fourth layer of the FNN were generated by general-izing those consequence parameters corresponding to each separate motion. Under this design, the number of FNN parameters needed to be managed was greatly reduced, because only the consequence parameters had to be dealt with. Simulation results show that this learning controller could successfully govern those motions in Fig. 8(a). We further applied the learn-ing controller to govern some test motions, shown in Fig. 9, that diEer from the motions in Fig. 8(a) in movement distance. In Fig. 9, the generated trajecto-ries approximate the reference ones quite well. The results demonstrate that this learning controller, with smaller memory allocation, was able to govern a group
of similar motions and generalize their corresponding FNN parameters to govern other similar motions.
Finally, in the third simulation, a group of arbi-trary motions with diEerent movement distances, ve-locities, loads, etc., were used for classi7cation, as shown in Fig. 10(a). These motions were divided into the six classes listed in Table 4, and are shown in Fig. 10(b): motions in classes 1–4 have similarities above 0.85, and motions in classes 5–6 have lower de-grees of similarity. The results show that motions with diEerent movement distances, velocities, and loads can be similar, and that motions with the same distances or loads may not be similar. This demonstrates that motion classi7cation based on using an FNN learning
Fig. 9. Test motion governing using a learning controller with smaller memory allocation.
Fig. 10. (a) A group of arbitrary motions. (b) Motions after classi7cation.
Table 3
The corresponding means and variances in the FNN for governing of motions with high degrees of similarity in Fig. 8(a) Input d1
Motion number Linguistic label
NB NS Z PS PB
Mean Variance Mean Variance Mean Variance Mean Variance Mean Variance
1 − 3:999 1:999 − 2:000 2:000 0:001 2:000 2:000 2:000 4:000 2:000 2 − 4:000 2:000 − 2:000 2:000 0:000 2:000 2:000 2:000 4:000 2:000 3 − 4:000 2:000 − 2:000 2:000 0:000 2:000 2:000 2:000 4:000 2:000 4 − 4:000 2:000 − 2:000 2:000 0:000 2:000 2:000 2:000 4:000 2:000 5 − 4:000 2:000 − 2:000 2:000 − 0:001 2:000 2:000 2:000 4:000 2:000 Average − 3:999 1:999 − 2:000 2:000 0:000 2:000 2:000 2:000 4:000 2:000 Input d2
Motion number Linguistic label
NB NS Z PS PB
Mean Variance Mean Variance Mean Variance Mean Variance Mean Variance
1 − 4:001 2:001 − 2:000 2:001 0:002 1:999 2:001 2:001 4:001 1:999 2 − 3:999 1:975 − 2:001 1:993 0:012 1:999 2:001 2:001 3:997 2:001 3 −3:998 2:001 − 2:002 2:004 − 0:002 2:001 2:001 2:001 3:989 2:003 4 − 4:000 2:001 − 1:999 2:001 −0:001 2:001 2:001 2:001 4:001 2:000 5 − 4:013 1:935 − 1:999 2:001 0:044 2:001 2:001 1:999 3:859 1:999 Average − 4:002 1:982 − 2:000 2:000 0:011 2:000 2:001 2:001 3:969 2:000 Input ˙d1
Motion number Linguistic label
S Z B
Mean Variance Mean Variance Mean Variance
1 − 2:996 2:596 − 0:045 2:531 2:596 3:640 2 − 2:946 2:595 − 0:032 2:483 2:645 3:620 3 − 2:999 2:499 0:001 2:496 2:732 3:212 4 − 2:999 2:499 − 0:001 2:499 2:844 2:956 5 − 2:804 2:721 − 0:004 2:534 2:915 2:935 Average − 2:948 2:582 − 0:016 2:508 2:746 3:272 Input ˙d2
Motion number Linguistic label
S B
Mean Variance Mean Variance
1 − 1:980 2:534 1:869 2:852 2 − 1:863 2:762 1:901 3:023 3 − 2:166 2:238 1:862 2:893 4 − 1:999 2:507 1:833 2:827 5 − 1:824 2:644 1:942 2:912 Average −1:966 2:537 1:881 2:901
Table 4 Motion index (( 1f ;2f ) Wn L ) after classi7cation in Fig. 10(b)
mechanism does not correspond only to the kinematic or dynamic features.
5. Discussion and conclusion
This paper has proposed motion similarity anal-ysis from the standpoint of learning. Similar mo-tions were de7ned as those corresponding to the same number of fuzzy rules and similar member-ship function shapes when governed using an FNN learning mechanism. By classifying motions ac-cording to their similarities, learning controllers can then be designed and allocated appropriate mem-ory sizes for motion governing. Simulations per-formed veri7ed the eEectiveness of the proposed approach.
At the current stage of the study, simulation and analysis are based on use of a two-joint robot ma-nipulator. As one of our future works, the proposed approach will be used to analyze motion similarities for general industrial robot manipulators. From our preliminary simulation results, we found when those motions, originally executed by two-joint robot ma-nipulators, were executed by six-joint robot manipu-lators, the fuzzy rule numbers in the rule layers of the six FNNs used for governing the six joints slightly increased from teens to still under 20. However, the time required for the FNN learning process much in-creased in tackling the highly complicated dynamic couplings present among joints. Because the fuzzy rule numbers and the corresponding membership function distributions are still manageable for similarity anal-ysis and motion classi7cation, we consider that the proposed approach is able to deal with general in-dustrial robot manipulators at the expense of learning time.
As the gradient-descent-based back-propagation algorithm is used for learning in the FNN, the learning may fall into diEerent local minima when the FNN learns to govern a motion. It means that the number of rules and shapes of mem-bership functions in the FNN may be diEerent in governing a motion, depending on the learn-ing result. Then, a question that may be raised is how the proposed motion similarity analysis and governing will be aEected under this situation. Although a motion may be categorized into
mo-tion classes with diEering similarities according to diEerent learning results, this motion still belongs to a class of motions with similar fuzzy parame-ters in their governing FNNs. Thus, we can still design a learning controller with smaller mem-ory allocation to perform the subsequent motion governing.
Because the motions used for simulations in this pa-per were generated by step signals, an issue of interest is how the system’s performance will be aEected when the proposed motion classi7cation scheme is applied to motions generated by diEerent kinds of input sig-nals. The simulation results show that when motions were generated by input signals in the form of steps, low-order polynomial functions, or sinusoids with dif-ferent frequencies, the FNN could successfully gov-ern the motions by using reasonable numbers of fuzzy rules, because these motions were in some sense well-behaved. We also let the FNN learn to guide the robot manipulator to track the trajectories of human hand-writing. The results show that the FNN required more fuzzy rules and learning time in handwriting trajec-tory tracking than in previous cases, due to the irregu-larity of the trajectory contours and the corresponding complicated dynamics.
A point that also deserves discussion is about the ef-fects of adopting diEerent types of FNNs or local con-trollers for the proposed motion classi7cation scheme. In the proposed scheme, the FNN and local controller were chosen from the current existing ones. It can be expected that when diEerent types of FNNs or local controllers were used for similarity analysis, the re-sulting analysis and subsequent motion classi7cation might be somewhat diEerent. However, we consider which types of FNNs or local controllers to be used in the proposed scheme may not be that crucial, because our intention is to show how to apply FNNs to im-plement the proposed similarity analysis and how to use local controllers to make fuzzy parameters in the FNN be more meaningful for similarity evaluation.
Finally, an interesting future work will focus on how to properly classify general robot motions over the entire learning space. Simulation results in Section 4 demonstrate that motion classi7cation via the means of learning does not correspond only to the kinematic or dynamic features, and proper features for motion classi7cation remain to be found. There-fore, classi7cation of general robot motions may
demand further investigation into similarities among the general motions, proper motion feature selection, and concomitant reorganization of the learning space. Appendix A: Description ofthe FNN
The FNN adopted for the proposed scheme consists of four-node layers, in which all nodes are of the same type within each layer [5]. Each of the four layers performs one stage of the fuzzy inference process, as described below:
Layer 1 (Input layer): Inputs in this layer are trans-mitted to the next layer directly without any compu-tation. As Fig. 4 shows, there are two nodes for two inputs r and ˙r for motions with a single degree of
freedom.
Layer 2 (Linguistic term layer): In this layer, crisp data are transformed into fuzzy data through linguistic labelling (small, large, etc.). Each node i in this layer has the node function
O2
i = Ai(x); (A.1)
where : X → [0; 1] is a membership function, x is the input to node i, and Aiis the linguistic label associated
with this node function. The Gaussian function is cho-sen for Ai(x), because it is a diEerentiable function
suitable for use in the learning process, described as Ai(x) = exp − x − ci ai 2 ; (A.2)
where aiand cirepresent the variance and mean of the
Gaussian function, respectively, and are referred to as premise parameters. DiEerent membership grades at the same crisp point can be obtained by adjusting the parameter set {ai; ci}. The Gaussian function is not the
only choice as the membership function; other contin-uous and piecewise diEerential functions can also be used.
Layer 3 (Rule layer): This layer is intended for implementation of the fuzzy rules. Each node in this layer corresponds to a rule, and has only one antecedent link from a linguistic-term node of a linguistic label to the output node. There are n = nr × n˙r rules in the FNN, where nr and
n˙r are the numbers of linguistic labels for inputs r and ˙r, respectively, and n the total number
of fuzzy rules. In this layer, each node outputs the 7ring strength of the rule, O3
i, by using the T-norm
operator to perform the fuzzy AND process [18]: O3
i = (x !
1+ x!2)1=!
2 ; (A.3)
where x1 and x2 are the outputs from the linguistic
term layer and !¿1. Note that there is no weight to be adjusted in this layer.
Layer 4 (Output layer): This layer is the output layer, and has as many nodes as there are output ac-tion variables. In Fig. 4, only one node is needed for a single motion command Cm. All consequence links
are fully connected to the output node, and each con-nected link has its own weight wi, referred to as the
consequence parameter. The defuzzi7cation approach adopted is the centroid defuzzi7cation method [9]: O4=n
i=1
wiOi3: (A.4)
The parameters to be learned in this FNN in-clude the premise parameters {ai; ci}, representing
the means and variances of the Gaussian function, and consequence parameters {wi}, representing the
weights of the consequence links. A gradient-descent-based back-propagation algorithm is employed to learn these parameters [13]. For the determination of the parameters via the learning process for generating motion commands Cm corresponding to input
mo-tions, an error rate is 7rst speci7ed in the last layer (Layer 4). This error rate is then back-propagated to adjust the parameters sequentially from layer to layer. Because a concise form of the inverse dynamic model of the robot manipulator is not available, the error rate cannot be obtained directly by diEerentiating the error between the desired motion and the actual mo-tion relative to the momo-tion command Cm. Instead, we
use the combined feedback error of position (e) and velocity (˙e) between the desired and actual motions, denoted as E = Gpe + Gd˙e, to derive the error rate
@E=@Cm[2,8]: @E @Cm = @E @O4 = (Gpe + Gd˙e); (A.5)
where is a learning rate and Gp and Gd are gains.
The error rate @E=@Cm in Eq. (A.5) is estimated, but
not exact, for describing the diEerential relationship between the motion command Cm and the resultant
motion. Nevertheless, the results in [2,8] and also ours show that the use of this error rate is appropriate for the learning. Using the error rate @E=@Cm and some
straightforward manipulation, we can derive updates for the parameters ai, ci, and wi.
Appendix B: Similarity measure oftwo fuzzy sets Suppose that A and B are two fuzzy sets with cor-responding centers, mA and mB, and widths, A and
B. The similarity measure between A and B can be
divided into four cases, as shown in Fig. 5. In Cases 2–4, it is assumed that mA¿mB; if mA¡mB, then mA
and mB, and A and B are switched.
Case 1: mA= mB and A¿ B. In this case, A and
B have the same center and no intersection point, as shown in Fig. 5(a). Using Eq. (2), the similarity mea-sure can be derived from
E(A; B) =M(B)M(A) = B
A; (B.1)
where
M(A ∩ B) = M(B); (B.2) M(A ∪ B) = M(A) + M(B) − M(A ∩ B) = M(A):
(B.3) From Eq. (B.1), the degree of similarity between A and B is equal to the ratio of B to A. In particular,
if A= B, then E(A; B) = 1; i.e., A = B.
Case 2: | A− B|√6mA−mB6( A+ B)√ and
mA¿mB. In this case, A and B have an intersection
point at (s1; h1), as shown in Fig. 5(b). From Fig. 5(b),
we can obtain M(A) + M(B) = ( A+ B)√; (B.4) M(A ∩ B) = 1 2(c1+ c2)h1; (B.5) where c1= A(mB− mA) + A( A+ B) √ A+ B ; (B.6)
c2= B(mB− mA) + B( A+ B) √ A+ B ; (B.7) h1= (mB− mA) + ( A+ B) √ ( A+ B)√ : (B.8)
From Eqs. (B.4) and (B.5), the similarity measure can be derived as
E(A; B) = (c1+ c2)h1
2( A+ B)√ − (c1+ c2)h1: (B.9)
Case 3: mA−mB6| B− A|√ and mA¿mB. In this
case, A and B have two intersection points at (s1; h1)
and (s2; h2), as shown in Fig. 5(c). There are two
conditions in this case: (i) A6 B and (ii) A¿ B.
From Fig. 5(c), we can obtain M(A ∩ B) =1 2[c1h1+ c2h2+ c3(h1+ h2)]: (B.10) For (i) A6 B: c1= A(mB− mA) + A( A+ B) √ A+ B ; (B.11) c2= A(mB− mA) + A( B− A) √ B− A ; (B.12) c3= 2 A√ − (c1+ c2); (B.13) h1= (mB− mA) + ( A+ B) √ ( A+ B)√ ; (B.14) h2= (mB− mA) + ( B− A) √ ( B− A)√ : (B.15) For (ii) A¿ B: c1= A(mB− mA) + B( A− B) √ B− A ; (B.16) c2= B(mA+ mB) + B( B− A) √ A+ B ; (B.17) c3= 2 B√ − (c1+ c2); (B.18) h1= (mB− mA) + ( A− B) √ ( A− B)√ ; (B.19) h2= (mB− mA) + ( A+ B) √ ( A+ B)√ : (B.20)
The similarity measure can be derived as E(A; B)
= 2( c1h1+ c2h2+ c3(h1+ h2)
A+ B)√ − [c1h1+ c2h2+ c3(h1+ h2)]:
(B.21) Case 4: mA − mB¿( A + B)√ and mA¿mB.
In this case, A and B have no intersection, as shown in Fig. 5(d). Thus,
M(A ∩ B) = 0: (B.22)
And the similarity measure is
E(A; B) = 0: (B.23)
References
[1] J.S. Albus, A new approach to manipulator control: the cerebellar model articulation controller (CMAC), ASME J. Dyn. Systems Meas. Control (1975) 220–227.
[2] S. Arimoto, Robustness of learning control for robot manipulators, IEEE Internat. Conf. Robotics Automat., 1990, pp. 1528–1533.
[3] H.R. Berenji, P. Khedkar, Learning and tuning fuzzy logic controllers through reinforcements, IEEE Trans. Neural Networks 3 (5) (1992) 724–740.
[4] C.T. Chao, Y.J. Chen, C.C. Teng, Simpli7cation of fuzzy-neural systems using similarity analysis, IEEE Trans. Systems Man Cybernet. Part B: Cybernet. 26 (2) (1996) 344–354.
[5] J.S.R. Jang, ANFIS: adaptive-network-based fuzzy inference system, IEEE Trans. Systems Man Cybernet. 23 (3) (1993) 665–685.
[6] S. Jung, T.C. Hsia, On reference trajectory modi7cation approach for cartesian space neural network control of robot manipulators, IEEE Internat. Conf. Robotics Automat., 1995, pp. 575–580.
[7] A. Karakasoglu, S.I. Sudharsanan, Identi7cation and decentralized adaptive control using dynamical neural networks with application to robotic manipulators, IEEE Trans. Neural Networks 4 (6) (1993) 919–930.
[8] M. Kawato, Y. Uno, M. Isobe, R. Suzuki, Hierarchical neural network model for voluntary movement with application to robotics, IEEE Control Systems Mag. 8 (2) (1988) 8–16. [9] C.C. Lee, Fuzzy logic in control systems: fuzzy logic
controller – Part I, IEEE Trans. Systems Man Cybernet. 20 (2) (1990) 404–418.
[10] C.M. Lim, T. Hiyama, Application of fuzzy logic control to a manipulator, IEEE Trans. Robotics Automat. 7 (5) (1991) 688–691.
[11] C.-T. Lin, A neural fuzzy control system with structure and parameter learning, Fuzzy Sets and Systems 70 (1995) 183–212.
[12] W.T. Miller III, F.H. Glanz, L.G. Kraft III, Application of a general learning algorithm to the control of robotic manipulators, Internat. J. Robotics Res. 6 (2) (1987) 84–98. [13] W.T. Miller III, R.S. Sutton, P.J. Werbos (Eds.), Neural Networks for Control, MIT Press, Cambridge, MA, 1990. [14] T. Ozaki, T. Suzuki, T. Furuhashi, S. Okuma, Y. Uchikawa,
Trajectory control of robotic manipulators using neural networks, IEEE Trans. Ind. Electron. 38 (3) (1991) 195–202.
[15] T.D. Sanger, Neural network learning control of robot manipulators using gradually increasing task diNculty, IEEE Trans. Robotics Automat. 10 (3) (1994) 323–333. [16] L. Sciavicco, B. Siciliano, Modeling and Control of Robot
Manipulators, McGraw-Hill, Singapore, 1996.
[17] T. Shibata, T. Fukuda, Hierarchical intelligent control for robotic motion, IEEE Trans. Neural Networks 5 (5) (1994) 823–832.
[18] C.T. Sun, Rule-base structure identi7cation in an adaptive-neural-based fuzzy inference system, IEEE Trans. Fuzzy Systems 2 (1) (1994) 64–73.
[19] M. Takegaki, S. Arimoto, A new feedback method for dynamic control of manipulators, ASME J. Dyn. Systems Meas. Control 103 (2) (1981) 119–125.
[20] B.A.M. Wakileh, K.F. Gill, Use of fuzzy logic in robotics, Comput. Ind. 10 (1988) 35–46.
[21] K.Y. Young, C.C. Fan, An approach to simplify the learning space for robot learning control, Fuzzy Sets and Systems 95 (1) (1998) 23–38.
[22] K.Y. Young, S.J. Shiah, An approach to enlarge learning space coverage for robot learning control, IEEE Trans. Fuzzy Systems 5 (4) (1997) 511–522.
