Robot learning schemes that trade motion accuracy for command
simpli cation
1
Kuu-young Young∗, Jyh-Fu Lee, Hui-Jun Jou
Department of Electrical and Control Engineering, National Chiao-Tung University, Hsinchu, Taiwan Received June 1997; received in revised form November 1997
Abstract
This study was inspired by the human motor control system in its ability to accommodate a wide variety of motions. By contrast, the biologically inspired robot learning controller usually encounters huge learning space problems in many practical applications. A hypothesis for the superiority of the human motor control system is that it may have simpli ed the motion command at the expense of motion accuracy. This tradeo provides an insight into how fast and simple control can be achieved when a robot task does not demand high accuracy. Two motion command simpli cation schemes are proposed in this paper based on the equilibrium-point hypothesis for human motion control. Investigation into the tradeo between motion accuracy and command simpli cation reported in this paper was conducted using robot manipulators to generate signatures. Signature generation involves fast handwriting, and handwriting is a human skill acquired via practice. Because humans learn how to sign their names after they learn how to write, in the second learning process, they somehow learn to trade motion accuracy for motion speed and command simplicity, since signatures are simpli ed forms of original handwriting. Experiments are reported that demonstrate the e ectiveness of the proposed schemes. c 2000 Elsevier Science B.V. All rights reserved.
Keywords: Robotics; Learning control; Command simpli cation; Human motor control; Fuzzy neural network
1. Introduction
Human limbs governed by the human motor con-trol system perform far better in many respects than those of robots, their industrial counterparts, a fact that has stimulated research into human limb motions and control strategies [4,5,9,15,17,20]. One appealing fea-ture demonstrated by the human motor control system is that it can accommodate a wide variety of motions through e ective memory management. By contrast, robot learning controllers, which are biologically
in-∗Corresponding author.
1This work was supported in part by the National Science Council, Taiwan, under grant NSC 84-2212-E-009-060.
spired and intended to emulate the human motor con-trol system, usually encounter huge learning space problems in many practical applications [16,18,21]. For example, using a learning controller, such as a neu-ral controller or a fuzzy system, to govern the geneneu-ral motion of a multi-joint robot manipulator demands quite a number of training patterns, and thus the re-sulting neural controller consists of a huge number of neurons, or the resulting fuzzy system requires numer-ous rules to govern motions [23]. This learning space problem severely hinders the application of learning control to robotics.
How the human motor control system resolves the aforementioned learning space problem is of in-terest. Intuition suggests that the superiority of the
0165-0114/00/$ - see front matter c 2000 Elsevier Science B.V. All rights reserved.
human motor control system can be attributed to its salient control strategies and exceptional ability to make proper decisions using information from abundant and versatile biological sensory feedback sources. On the other hand, it has been observed that human limb motions are not very accurate, leading to the hypothesis that the human motor control system may have simpli ed its learning space at the expense of motion accuracy [22]. Although there is still doubt over whether the human motor control system actually performs this tradeo , the hypothesis is not based on weak evidence. Results on the speed-accuracy tradeo in human movements and others have been reported in several human motor control studies [17]. Inspired by this hypothesis, according to the degree of accuracy given up, motion control can be transitioned from ac-curate motion tracking toward point-to-point motion regulation via learning. Consequently, the original complex motion commands capable of tracking the motion accurately are simpli ed. With motion com-mands in simple forms, learning controllers can then be designed that do not consume excessive memory resources. In addition, simpli ed motion commands also lead to fast and simple command execution and smooth motion control with fewer oscillations.
Investigation into the tradeo between motion ac-curacy and command simpli cation reported in this paper was conducted using robot manipulators to generate signatures. Signatures are usually generated rapidly and with little demand for motion accuracy, yet handwriting is a skilled human action [14]. The skills of handwriting and signature generation are both acquired via learning; humans learn how to sign their names after they learn how to write. In other words, humans learn to achieve simplicity in writing a signature by giving up certain degree of accuracy after they have learned how to accurately approximate the handwriting. With this appealing feature, signa-ture generation stands as an excellent example that suits our purpose. To implement a similar tradeo , two command simpli cation schemes are proposed based on the equilibrium-point hypothesis, discussed in Section 2 [17]. The rest of the paper is organized as follows. In Section 2, biological backgrounds related to the human motor control system and equilibrium-point hypothesis are presented. In Section 3, handwriting generation processes and schemes are described. The proposed command simpli cation
schemes are discussed in Section 4. In Section 5, ex-perimental results and analyses are reported. Finally, conclusions are given in Section 6.
2. Human motor control system and equilibrium-point hypothesis
Fig. 1 shows a simpli ed block diagram of the hu-man motor control system that governs limb motion. In Fig. 1, we can see that human motion is governed by a hierarchical structure [9,10,17]. In response to various demands, the central nervous system (CNS) makes motion plans. Appropriate motor commands are then generated and sent to the peripheral neu-romotor system, which may then modify the motor commands according to sensory feedback. The periph-eral neuromotor system behaves as a local controller that adapts to di erent motions, loads, and environ-ments, in addition to accepting commands from the CNS. Finally, the modi ed commands are sent to the muscular-skeletal system for motion execution. With this hierarchical structure, the diculty of perform-ing complex motions can be shared by the CNS at the higher level and the local controller at the lower level. Among those hypotheses for human motion con-trol, the equilibrium-point hypothesis suggests that the CNS speci es equilibrium points between agonist and antagonist muscle groups that correctly posi-tion limbs in relaposi-tion to the target by indicating new sets of length-tension curves for the muscle groups [4,9,17]. In other words, motions are treated as tran-sitions between postures. The CNS needs only select new levels for the motor commands. The subsequent result, mediated by autogenetic re exes and the me-chanical properties of the muscles, should be a smooth transition from one posture to another. The simple control-signal format makes the equilibrium-point hypothesis very attractive for robot motion control, although there are still debates and controversies in this hypothesis. However, since only one equilibrium point is selected, a control strategy based on this hy-pothesis would not enable us even to vary the motion speed between two given postures. To exploit the simplicity of the equilibrium-point hypothesis and enable it to deal with di erent velocities and loads in reaching various positions, motor commands may consist of numbers of equilibrium points. Thus, slow
Fig. 1. A simpli ed block diagram of the human motor control system.
motions can be produced by progressive shifts of equilibrium points. Motions can be speeded up by assigning an initial shift that is larger than necessary, followed by a return to a proper static level [9]. In light of both physiological and engineering consider-ations, the number of equilibrium points in the motor command should be kept fairly small [5,11,20].
The proposed motion command simpli cation schemes were developed according to the equilibrium-point hypothesis. By applying the proposed schemes, the original complex motion commands for motion governance can be simpli ed into series of square pulses of various heights and widths [11,20,22]. With the controlled parameters in the motion command being the heights and widths of the square pulses, the learning space for dealing with variations exhibited by di erent motions is dramatically reduced. This motion command simpli cation is, however, achieved by sacri cing motion accuracy, because continuous control signals suitable for accurate tracking are ap-proximated by signals consisting of square pulses. Note that the controlled parameter in the equilibrium-point hypothesis is muscle compliance instead of the equilibrium point used in the proposed schemes. Our purpose is not to propose a new biological hy-pothesis, but to develop control strategies for robot motion control inspired by the human motor control system.
3. Handwriting generation
Before the proposed motion command simpli ca-tion schemes can be applied to robot manipulators for
signature generation, samples of the handwriting the signature is derived from need to be provided rst. In Section 3.1, handwriting generation processes are introduced along with a survey of previous hand-writing generation schemes. A handhand-writing learning scheme (HLS) is then discussed in Section 3.2 to derive motion commands capable of tracking the handwriting for the robot manipulator. With the hand-writing and its corresponding continuous, complex motion commands available, the tradeo between motion accuracy and command simpli cation can be demonstrated via teaching robot manipulators to generate signatures.
3.1. Previous works
Fig. 2 shows a typical handwriting process, in-cluding four stages: cognitive decision, trajectory formation, motor command generation, and motion execution [14]. In Fig. 2, the linguistic information is rst transformed into a stream of words. Because the shapes of words are usually complex, they are divided into letters and then strokes during trajectory formation. The trajectory of each stroke is planned according to various considerations, such as size, shape, speed, and location. Various criteria have been proposed for trajectory planning to accom-modate di erent design purposes, such as energy conservation, maintenance of the bell-shaped veloc-ity pro le, and trajectory smoothness [3,7]. In the next stage, the CNS generates motor commands to realize the planned trajectories by using proper con-trol parameters. Finally, the muscular system accepts
Fig. 2. The handwriting process.
commands from the CNS to execute handwriting mo-tions. Note that the cognitive aspect of handwriting will not be dealt with in this paper, since the study is not intended for how various handwriting to be generated.
Various handwriting generation models have been proposed and can be divided into two major classes: the muscle-oriented model and the space-oriented model [13]. In the muscle-oriented model, trajectory generation is directly related to the con gurations of the muscles and their mechanical properties. Most of the models reduce the complexity of the biomechan-ical system for handwriting by factoring it into two orthogonal functional degrees of freedom. It is gener-ally assumed that horizontal movements are produced by rotation of the hand about the wrist, and vertical movements by oscillations of the thumb, index nger, and middle nger. Mathematical equations describing muscle dynamics are then used to generate
hand-writing according to di erent types of input stimuli. Hollerbach proposed an oscillation model for control-ling the shape, height, and slant of handwriting [7]. Two orthogonal pairs of springs were used to generate the required oscillations for handwriting. Plamondon and Maarse proposed a more general model to de-scribe and analyze biomechanical handwriting [14]. In their model, a second-order sub-system was used to represent the hand-pen-paper system and a rst-order one used as a nerve-muscle interface.
In the space-oriented model, trajectory generation is based on an ability to express and control the trajectory of the hand in space, independent of the actual joint and muscle system. The model is sup-ported by the fact that humans can write in the same way on a sheet of paper and on a blackboard, using the hand or even other parts of the body. Morasso and Ivaldi proposed a trajectory formation model for handwriting [13]. In their model, handwriting was produced by a mechanism of composition of discrete strokes represented by a weighted sum of B-splines. The mechanism was also able to generate smooth tra-jectories by means of timed overlapping of di erent strokes. Edelman and Flash proposed a handwrit-ing generation model based on the kinematics from shape principle and on dynamic optimization [3]. Symbolic descriptions of strokes were used in their model.
3.2. Handwriting learning scheme (HLS)
Fig. 3 shows a block diagram of the proposed hand-writing learning scheme based on using the two-joint planar robot manipulator shown in Fig. 4(a). Because the cognitive aspect of handwriting is beyond the scope of this paper, the process of transforming mes-sages into chains of strokes is ignored in our scheme. In Fig. 3, human subjects rst input samples of their handwriting by writing on a digital tablet. Input hand-writing samples are then transformed into Cartesian trajectories (Pd(t); Vd(t)) in the robot workspace
ac-cording to the coordinate system of the human hand (as determined through the digital tablet) via a tra-jectory mapping process. The Cartesian tratra-jectory (Pd(t); Vd(t)) is further mapped into a joint trajectory
(qd(t); ˙qd(t)) via an inverse-kinematic
transforma-tion. According to the joint trajectory (qd(t); ˙qd(t)), a
Fig. 3. The handwriting learning scheme (HLS).
Fig. 4. (a) A two-joint planar robot manipulator. (b) Robot workspace partition.
Fig. 2, generates motion commands EP(t) for tra-jectory tracking. Note that motion commands gen-erated by the FNN consist of equilibrium points in continuous form. In turn, a local controller, that
emulates the peripheral neuromotor system in the muscular system shown in Fig. 2, modulates the motion commands via sensory feedback and uses the resultant torque (t) to move the robot and pen system.
According to some biological evidence, the CNS may provide only the desired equilibrium points for motion control [15]. Therefore, to simplify the design of our scheme, only the desired equilibrium points and no desired velocities are speci ed in the motion commands. A simple position control law with linear damping is then used for the local controller [19]:
= Kp(EP − q) − Kd˙q; (1)
where EP stands for the equilibrium point vector, q and ˙q are the actual position and velocity vectors obtained via sensory feedback, and Kp and Kd are
symmetric-positive-de nite matrices for stability con-siderations [2].
In the proposed scheme, trajectory mapping is per-formed from the human hand coordinate system to that of the two-joint planar robot manipulator, because the human hand and the two-joint robot manipulator are di erent mechanisms with di erent kinematic and dy-namic features, and thus they choose di erent optimal locations in their own workspaces and use di erent con gurations to better handwriting generation. Mean-while, when the handwriting is placed at di erent lo-cations in the workspace for the robot manipulator to track, di erent learning results, consequently di erent equilibrium points, are obtained. To nd optimal loca-tions at which to place the handwriting, we performed a series of simulations using the proposed HLS and adopted the minimum-equilibrium-point-change crite-rion proposed in [6] for performance evaluation. This
Fig. 5. The structure of the FNN.
criterion aims for a smooth transition between pos-tures. Fig. 4(b) shows the robot workspace in the rst quadrant partitioned into nine regions; several loca-tions in each region were chosen for evaluation. Sim-ulation results showed that when the handwriting was placed at locations chosen from the gray region in the middle portion, shown in Fig. 4(b), the derived equi-librium points for trajectory tracking most satis ed the criterion, and this region in the robot workspace was then used as the handwriting location.
The FNN for motion command generation is basi-cally a fuzzy system implemented in the form of a
neural network, as shown in Fig. 5 [1,12]. The rep-resentation of a fuzzy system using a fuzzy neural network enables us to take advantage of the learning ability of the neural network for automatic tuning of the parameters in the fuzzy system. The fuzzy reason-ing parameters are thus expressed in the connection weights or node functions of the neural network. In the proposed HLS, we chose an FNN with a structure similar to that in [1]; of course, other types of FNN can also be used. As Fig. 5 shows, the inputs to the FNN are the joint position and velocity trajectories of the input motions, (qd(t); ˙qd(t)), and the outputs are the
equilibrium point trajectories EP(t). We assume that stability and convergence of the FNN in learning to track continuous trajectories are guaranteed, and these issues are well dealt with in previous studies [1,12]. Our previous results have demonstrated that the FNN is capable of governing continuous robot trajectories [23], and the results in this paper also show that the handwriting generated by the HLS approximated the originals quite well. Detailed discussions of the struc-ture and learning process of this FNN can be found in the appendix.
4. Proposed command simpli cation schemes Two schemes are proposed to implement the tradeo between motion accuracy and command simpli cation. Fig. 6 shows the gradual learning sim-pli cation scheme (GLSS) and the command shape simpli cation scheme (CSSS). These two schemes are applied to simplify motion commands for hand-writing generation into those for signature generation. Correspondingly, the continuous equilibrium point trajectories derived by the HLS in Section 3.2 are simpli ed into trajectories consisting of series of square pulses of various heights and widths. During the simpli cation process, the schemes gradually give up accuracy in approximating the handwriting tra-jectory, and the resulting handwriting becomes more and more like actual signatures.
4.1. Gradual learning simpli cation scheme (GLSS) Fig. 6(a) shows a block diagram of the gradual learning simpli cation scheme (GLSS). In the GLSS, tradeo between motion accuracy and command sim-pli cation is achieved via a simsim-pli cation process in-volving the error evaluator, the updating gate, and the FNN, as shown in the blocks surrounded by the dot-ted lines in Fig. 6(a). The FNN used in the GLSS is the same as that used in the HLS in Section 3.2 with the learning process for handwriting tracking in the HLS completed. Therefore, before the simpli cation process in the GLSS is executed, the input joint tra-jectory (qd(t); ˙qd(t)) will elicit from the FNN
contin-uous equilibrium point trajectories EP(t) able to track the handwriting accurately. In the GLSS simpli cation process, an error bound is rst set in the error
evalua-tor. This error bound indicates the amount of accuracy to be traded for command simpli cation for a portion of the EP(t). The design will make the tradeo be per-formed in each local portion, leading to a more homo-geneous tradeo over the entire trajectory. When the cumulative error in tracking the handwriting does not exceed the error bound, the updating gate is closed, preventing the FNN from continuing to update mo-tion commands. Thus, the momo-tion commands remain at xed values during that period. Consequently, the resulting EP(t) will be in the form of series of square pulses. By contrast, a general learning mechanism, in some sense, has the error bound set to zero, and thus updates itself at every sampling time, resulting in con-tinuous motion commands.
The number of square pulses in EP(t) after com-mand simpli cation depends on the value of the error bound: when the error bound is large (small), EP(t) will have a small (large) number of square pulses. We use the joint velocity error for the er-ror bound, because variations in joint velocities are more evident than those in joint positions. The er-ror bound is de ned as the sum of the joint one and two velocity errors, since the scales of velocity error variations of joints one and two are similar ac-cording to our observations. The motion commands for both joints will then be updated simultaneously each time the error bound is exceeded. We set the error bound to a small value at the beginning of the simpli cation process and increase it gradually. Intuition suggests that the nal value of the error bound can be determined according to the preset similarity criterion between the original handwriting and the resulting signature. However, the resem-blance between these two is quite subjective and qualitative. In order to quantitatively describe the similarity between the handwriting and the signa-ture, we propose the concept of similarity bounding. The similarity bound Es is de ned as several times
the total Cartesian error Ec between the input
hand-writing after trajectory mapping in the HLS, Ti(t),
and handwriting learning by the HLS, Th(t), as
follows:
Es = kEc; (2)
where k¿1 is an empirical value, standing for the degree of similarity and referred to as a similarity
Fig. 6. (a) The gradual learning simpli cation scheme (GLSS). (b) The command shape simpli cation scheme (CSSS).
index. A proper selection of k should make the orig-inal handwriting still recognizable from the resulting signature. The total Cartesian error Ec can be
com-puted using the following equation: Ec=
n
X
i=1
[(xd(i) − xh(i))2+ (yd(i) − yh(i))2]1=2; (3)
where (xd; yd) and (xh; yh) are the coordinates of the
samples of Ti(t) and Th(t), respectively, and n is the
number of samples.
Based on the discussions above, the command sim-pli cation in the GLSS will begin with a small initial error bound. The joint velocity error will be evaluated at each sampling time. The FNN will update motion commands only when the cumulative error exceeds the error bound. After all handwriting command sim-pli cation has been completed, the total Cartesian er-ror E between Ti(t) and the resulting trajectory Tl(t)
is computed. When E is smaller than the similar-ity bound Es, the error bound will be increased and
command simpli cation resumed. The simpli cation
process will proceed until E is greater than Es. To
summarize, the algorithm for the operation in the GLSS is:
Gradual Learning Simpli cation Algorithm: Sim-plify continuous equilibrium point trajectories into trajectories consisting of series of square pulses via a simpli cation process with feedback for evalua-tion using pre-speci ed degrees of similarity between originals and derived trajectories.
Step 1: Input Ti(t), Th(t), the velocity trajectory
Vh(t), and the equilibrium point trajectory EPh(t)
cor-responding to Th(t).
Step 2: Compute the total Cartesian error Ec
be-tween Ti(t) and Th(t). Determine the similarity bound
Esby selecting an empirical similarity index k.
Step 3: Initialize the error bound with a small value. Step 4: Evaluate the joint velocity error between the current joint velocity and the reference joint velocity corresponding to Vh(t) at each sampling time. Allow
the FNN update motion commands only when the cu-mulative joint velocity error exceeds the error bound.
Step 5: Compute the total Cartesian error E be-tween Ti(t) and Tl(t) after all handwriting command
simpli cation.
Step 6: Check whether E is smaller than Es; if yes,
increase the error bound and go to Step 4; otherwise, the simpli cation process is completed and output the simpli ed equilibrium point trajectory EPs(t) as series
of square pulses.
4.1.1. Command scaling
The GLSS can also be used to trade motion ac-curacy for simpli ed motion commands that generate motions similar to the original motion, but di erent in movement distance and velocity. By performing motion command simpli cation and scaling simulta-neously, the GLSS is able to achieve motion command scaling without system dynamics recalculation [8]. Possible industrial applications for this feature of the GLSS can be providing simpli ed motion commands for industrial tasks that involve a number of similar robot motions with di erent movement distances and velocities.
In the application of signature generation, the equi-librium point trajectory EPh(t) corresponding to the
handwriting Th(t) is simpli ed and scaled by the
GLSS to generate signatures of di erent sizes with di erent velocities. This function of the GLSS can be achieved following the Gradual Learning Simpli ca-tion algorithm above with modi caSimpli ca-tion of the error evaluation process in Steps 4 and 5. In the new eval-uation process, Th(t) and its velocity trajectory Vh(t)
are scaled and used for reference. The error evaluator will then compare the trajectories generated by the algorithm with the reference trajectories after scaling, simplifying and scaling EPh(t) simultaneously. The
error bound may also need to be increased (decreased) according to di erent size and velocity requirements. To generate signatures of di erent sizes, Th(t) and
Vh(t) can be scaled by varying the sampling time, as
follows:
r = ct; (4)
ˆqh(r) = (1 − c) × qh0+ c × qh(t); (5)
˙ˆqh(r) = ˙qh(t); (6)
where c is a scaling constant, (qh(t); ˙qh(t)) are the
joint position and velocity trajectories corresponding
to Th(t) and Vh(t), respectively, qh0is the initial joint
position of qh(t), and ( ˆqh(r); ˙ˆqh(r)) are the scaled joint
position and velocity trajectories, respectively. When c ¿ 1, it is ampli cation, and vice versa. To gener-ate signatures with di erent velocities, scaling is im-posed upon both the sampling time and the velocity, as follows:
r = ct; (7)
ˆqh(r) = qh(t); (8)
˙ˆqh(r) = ˙qh(t)
c : (9)
When c¡1, it is a speed-up, and vice versa.
4.2. Command shape simpli cation scheme (CSSS) Fig. 6(b) shows a block diagram of the command shape simpli cation scheme (CSSS). In the CSSS, the tradeo between motion accuracy and command simpli cation is performed according to the charac-teristics of the command shapes. Local extreme val-ues on the equilibrium point trajectories EPh(t) are
rst located and EPh(t) are then approximated using
a series of zero-order polynomials, which replace the curves (or lines) between local extreme points with square pulses. When the di erence between two ad-jacent pulses in the approximated EPh(t) is smaller
than some pre-speci ed threshold in either amplitude or duration, the pulses will be combined via process-ing in command amplitude or duration, as shown in the blocks surrounded by the dotted lines in Fig. 6(b). Note that the amplitude and duration thresholds for the two joints of the robot manipulator may be cho-sen di erently according to variations in the command shapes for each joint. The nal similarity between the original handwriting and the resulting signature in the CSSS is also speci ed using the similarity bound, as in the GLSS. The amplitude and duration thresholds will be initialized with small values and be increased gradually when the total Cartesian error E between Ti(t) and the resulting trajectory Tl(t) after command
simpli cation does not exceed the similarity bound. To summarize, the algorithm for the operation in the CSSS is:
Command Shape Simpli cation Algorithm: Simplify continuous equilibrium point trajectories into trajecto-ries consisting of setrajecto-ries of square pulses according to the command shape characteristics using pre-speci ed degrees of similarity between originals and derived trajectories.
Step 1: Input Ti(t), Th(t), and the equilibrium point
trajectory EPh(t) corresponding to Th(t).
Step 2: Compute the total Cartesian error Ec
be-tween Ti(t) and Th(t). Determine the similarity bound
Esby selecting an empirical similarity index k.
Step 3: Initialize the amplitude and duration thresh-olds with small values.
Step 4: Locate local extreme values on EPh(t). Use
zero-order polynomials to approximate EPh(t) by
re-placing the curves (or lines) between local extreme points with square pulses.
Step 5: Perform command amplitude and duration processing to combine motion commands for the ap-proximated EPh(t).
Step 6: Compute the total Cartesian error E between Ti(t) and Tl(t) after command combination for the
entire approximated EPh(t) is completed.
Step 7: Check whether E is smaller than Es; if yes,
increase the amplitude and duration thresholds and go to Step 5; otherwise, the simpli cation process is completed and output the simpli ed equilibrium point trajectory EPs(t) as series of square pulses.
By applying this algorithm, the nal simpli ed equi-librium point trajectories EPs(t) will be a series of
square pulses. To smooth the EPs(t), we can include
a command smoothing process that approximates the square pulses of the EPs(t) using second-order
polyno-mials. Obviously, other kinds of functions, e.g., spline functions, can also be used for approximation. 5. Result and analysis
To demonstrate the e ectiveness of the two pro-posed command simpli cation schemes, the GLSS and the CSSS, they were applied to simplify the equi-librium point trajectories for handwriting generation into those for signature generation for the two-joint planar robot manipulator, shown in Fig. 4(a). Three adult subjects, two male and one female, were asked to provide handwriting samples. They practiced
writ-ing on the digital tablet for a while, and their sam-ples were recorded after they were con dent about using the digital tablet. The subjects were told to write quickly to generate more natural handwriting, and to select satisfactory samples from what they wrote according to their own standards. The selected sam-ples were then mapped into Cartesian trajectories Ti(t) in the robot workspace using the HLS described
in Section 3.2. Via a learning process in the HLS, the equilibrium point trajectories EPh(t) were derived,
which in turn generated trajectories Th(t)
approxi-mating Ti(t).
The two-joint planar robot manipulator was used to simulate the hand and pen system, and its dynamic equations are expressed as follows:
1 2 = H11 H12 H21 H22 1 2 + " −c ˙22− 2c ˙1˙2 c ˙21 # ; (10) where H11= m1l2c1+ m2l21+ m2l2c2 +2m2l1lc2cos (2) + I1+ I2; (11) H12= m2l2c2+ m2l1lc2cos (2) + I2; (12) H21= H12; (13) H22= m2l2c2+ I2; (14) c = m2l1lc2sin (2); (15)
with 1 and 2 standing for the torques, 1 and
2 the joint variables, m1= 2:815 kg, m2= 1:64 kg,
l1= 0:3 m, l2= 0:32 m, lc1= 0:15 m, lc2= 0:16 m,
and I1= I2= 0:0234 kgm2. The e ects of load and
gravity were ignored in the formulation, and the sampling time in the simulation was 2 ms. For all schemes, the HLS, the GLSS, and the CSSS, each joint of the robot manipulator was equipped with an FNN and a local controller. In each FNN, there were two nodes in Layer 1, ten nodes in Layer 2, 25 nodes in Layers 3 and 4, and one node in Layer 5. The local controller gains were set to Kp= 15 and 10 N m=rad
and Kd= 3 and 1 N m=(rad=s) for joints one and two,
Fig. 8. Motion command simpli cation for the name ‘Chen’: (a) the HLS, (b) the GLSS, (c) the CSSS, and (d) the CSSS plus smoothing.
Fig. 7 shows the resulting position trajectories and the corresponding equilibrium point trajectories for an input handwritten character ‘a’ from (a) the HLS, (b) the GLSS, and (c) the CSSS. In Fig. 7(a), the total Cartesian error Ec between the input
handwrit-ten ‘a’ trajectory used for reference (dotted line) and the trajectory generated by the HLS (solid line) was computed to be about 0.1 m. The equilibrium point trajectories EPh(t) derived by the HLS were
con-tinuous, and were sent to the GLSS and the CSSS for command simpli cation. Figs. 7(b) and (c) show the EPh(t) from Fig. 7(a) simpli ed into series of
square-pulse trajectories. For both the GLSS and the
CSSS, the similarity index k was set to 5, making the similarity bound equal to 0.5 m. The total Cartesian errors E between the reference and generated trajec-tories after command simpli cation were about 0.53 and 0.59 m for the GLSS and the CSSS, respectively. From the results, both the GLSS and the CSSS can generate simpli ed motion commands that result in pre-speci ed degrees of similarity between the origi-nal and the derived trajectories. In general, the GLSS can generate more accurate trajectories using the same similarity bound, but consumes more computa-tion time, as compared to the CSSS. This is because in command simpli cation, the GLSS uses a
simpli-Fig. 8. Continued.
cation process with feedback for evaluation, while the CSSS performs command combination directly on the command shapes.
In the second case study, we used a more compli-cated sample, the name ‘Chen’, and also evaluated the e ect of the CSSS when the command smoothing process described in Section 4.2 was included. Fig. 8 shows the resulting position trajectories and the corresponding equilibrium point trajectories for the input handwritten sample of the name ‘Chen’ from (a) the HLS, (b) the GLSS, (c) the CSSS, and (d) the CSSS plus smoothing. In Fig. 8(a), Ecafter learning
was computed to be about 0.1 m. For both the GLSS and the CSSS, the similarity index k was set to 5. In Figs. 8(b) and (c), the EPh(t) in Fig. 8(a) derived by
the HLS were simpli ed into series of square-pulse
trajectories. The number of square pulses for the handwritten ‘Chen’ was greater than that for ‘a’ as expected. The total Cartesian errors E after command simpli cation were about 0.53 and 0.57 m for the GLSS and the CSSS, respectively. Fig. 8(d) shows the result when the command smoothing process was included in the CSSS. Second-order polynomials were used to approximate the motion command trajectories in Fig. 8(c). In Fig. 8(d), the resulting motion com-mand trajectories were smooth and the total Carte-sian error E after command smoothing was about 0.55 m, demonstrating the feasibility of the proposed command smoothing technique.
Finally, in the third case study, we evaluated the performance of applying the GLSS with com-mand scaling, as described in Section 4.1.1, to
Fig. 9. Generation of the character ‘h’ under di erent size and velocity requirements using the GLSS: (a) the reference ‘h’, (b) a larger ‘h’, and (c) a faster ‘h’.
generate signatures of di erent sizes and velocities. Fig. 9(a) shows the character ‘h’ used for refer-ence, generated by the HLS with Ec after
learn-ing about 0.1 m, and the correspondlearn-ing continuous equilibrium point trajectories EPh(t). Command
scaling was applied to generate a larger ‘h’ and a normal ‘h’ written more rapidly. The reference tra-jectories for error evaluation during command scaling were generated using Eqs. (4)–(9), with the scaling constants c = 1:5 and 1.25 for the larger ‘h’ and the faster ‘h’, respectively. Due to the increases in size and writing velocity, the similarity indices were in-creased accordingly and set to 26 and 8 for the larger ‘h’ and the faster ‘h’, respectively. In Figs. 9(b) and (c), the EPh(t) in Fig. 9(a) were simpli ed and scaled
into series of square-pulse trajectories, which were able to generate larger and faster ‘h’s with errors within the similarity bounds. This demonstrates the feasibility of the proposed command scaling. 6. Conclusion
In this paper, we have developed motion command simpli cation schemes that can trade motion accuracy for command simpli cation in robot motion control. The proposed command simpli cation is taken as a second learning process after accurate motion track-ing that demands complicated motion commands has been accomplished. Thus, the proposed schemes pro-vide e ective frameworks for achieving fast, simple control when a task does not demand high accuracy, and to transition between motion tracking and regula-tion according to the degree of moregula-tion accuracy given up. The results of applying the proposed schemes to simplify motion commands for handwriting genera-tion into those for signature generagenera-tion demonstrate the e ectiveness of the proposed schemes. In future works, the proposed schemes will be applied to gen-eral industrial robot tasks, and to the search for sim-ple, basic motion commands that capture fundamental motion features.
Appendix. Description of the FNN
The structure of the FNN used in the proposed schemes consists of ve layers of nodes, all of which
are of the same types within the same layer, as shown in Fig. 5. Each of the ve layers performs one stage of the fuzzy inference process, as described below:
Layer 1. The input layer: It transmits inputs directly to the next layer without performing any computation. As Fig. 5 shows, there are two nodes for two inputs qd
and ˙qd for motions with a single degree-of-freedom. Layer 2. The input membership function layer: It transforms input data into fuzzy data. Each node i in this layer has the node function
O2
i = (x); (A.1)
where : X → [0; 1] a membership function and x is the input to node i. The triangular membership func-tion adopted is described below:
(x) = 1 − (x − b)c ; x ∈ [b; b + c]; 1 + (x − b)a ; x ∈ [b − a; b]; 0 otherwise: (A.2)
Di erent membership grades at the same crisp point can be obtained by adjusting the parameter set (a; b; c). Layer 3. The rule layer: It implements fuzzy rules. Each node in this layer corresponds to a rule, de ned as a fuzzy conditional statement of the form
Rule: IF X is A and Y is B THEN Z is C; (A.3) where X and Y are fuzzy sets representing the inputs, Z represents the output, and A, B, and C represent lin-guistic variables, such as small, medium, and large. The number of rules involved in the input–output rela-tionship is pre-speci ed. In this layer, each node also outputs the ring strength of the rule, O3
i, by
perform-ing the di erentiable softmin operation [1]: O3 i = P jOj2exp(−rOj2) P jexp(−rOj2) ; (A.4) where O2
j is the output of the jth node in Layer 2
con-nected to the ith node in Layer 3 and r is a constant. When r approaches in nity, the softmin operator be-comes a min operator; for nite r, O3
i is di erentiable,
which is required during the learning process. Layer 4. The output membership function layer: Each node i in this layer performs an inversion of
i to locate the X -coordinate of the centroid of the
membership function, O4
i, using the local
mean-of-maximum method (LMOM) [1]: O4
i= −1i (O3i): (A.5)
Layer 5. The output layer: It has as many nodes as there are output action variables. Fig. 5 shows only one node is needed for the single motion command EP. The defuzzi cation approach adopted is the weighted averaging method:
O5=PPiO3iO4i
iO3i : (A.6)
Because the number of rules in Layer 3 is pre-speci ed and weights for the input and output layers (Layers 1 and 5) are xed, the parameters to be learned in this FNN are the modi able weights present at the input links to Layers 2 and 4, which correspond to the input and output membership functions. When the FNN learns the input and output membership function parameters required to generate the motion command EP corresponding to a sampled motion, an error rate, related to the motion command EP and the resultant motion, is initially speci ed in the last layer (Layer 5). This error rate is then back-propagated to adjust the parameters from layer to layer sequentially. Because a concise form of the inverse dynamic model of the robot manipulator is not available, the error rate cannot be obtained directly by di erentiating the er-ror between the desired motion and the actual motion relative to the motion command. 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=@EP
[10]: @E @EP= @E @O5 = (Gpe + Gd˙e); (A.7)
where is a learning rate and Gp and Gd are gains.
The error rate @E=@EP in Eq. (A.7) is estimated, but not exact, for describing the di erential relationship between the motion command EP and the resultant motion. Nevertheless, the results in [10] and also ours show that the use of this error rate is appropriate for the learning. Using the error rate @E=@EP and some
straightforward manipulation, we are able to derive updates for the parameters in Layers 2 and 4. References
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