near-minimum-time task planning for fruit-picking robots

48 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 7, NO. 1, FEBRUARY 1991

Near-Minimum-Time Task Planning for Fruit-Picking Robots

Yael Edan, Tamar Flash, Uri M. Peiper, Itzhak Shmulevich, and Yoav Sarig

Abstract-A near-minimum-time task-planning algorithm for fruit- harvesting robots having to pick fruits at N given locations is presented. For the given kinematic and inertial parameters of the manipulator, the algorithm determines the near-optimal sequence of fruit locations through which the arm should pass and finds the near-minimum-time path between these points. The sequence of motions was obtained by solving the Traveling Salesman Problem (TSP) using the distance along the geodesics in the manipulator’s inertia space, between every two fruit locations, as the cost to be minimized. The algorithm presented here was applied to define the motions of a citrus-picking robot and was tested for a cylindrical robot on fruit position data collected from 20 trees. Significant reduction in the required computing time was achieved by dividing the volume containing the fruits into subvolumes and estimating the geodesic distance rather than calculating it. Nevertheless, in most cases the solution of the TSP, based on the estimated geodesic distance, produced nearly the same fruit sequence as the one resulting from the use of the exact geodesic distance between the fruit locations. Results of simulation tests enabled us to assess the influence of the robot’s kinematic and dynamic parameters and of the spatial distribution of fruits on the motion sequence being selected. The proposed algorithm can help in selecting the most efficient robot design for any robot having to perform a sequence of tasks at N known locations.

I. INTRODUCTION 0 increase productivity is an important goal in industrial T automation. The planning of minimum-time movements for

a robot manipulator is one way to achieve this goal. Given a set of N tasks to be performed at known locations, it is necessary to determine the minimum-time path of a robotic arm passing once and only once through each task point. The main objective of this work is to suggest a method for reducing the cycle time required for the robot manipulator to complete a set of given tasks by developing a near-minimum-time task-planning algorithm. This algorithm enables us to determine the near-optimal sequence of task points through which the arm should pass and to find the minimum-time path between these points. Previously existing algorithms have dealt with the problem of finding the minimum-time path between two points. Due to the nonlinearity and closely coupled nature of the manipulator’s equations of motions, this is a rather complicated problem to solve. Hence, this problem is usually divided into two subproblems: trajectory planning and trajectory tracking.

The trajectory-planning problem deals with calculating the time history of joint positions, velocities, accelerations, and

Manuscript received February 21, 1989; revised June 22, 1990. This work was presented io part at the Agricultural Engineering International Conference, Paris, March 2-5, 1988.

Y. Edan was with the Agricultural Engineering Institute, Agricultural Research Organization, P.O.B. 6, Bet-Dagan 50250, Israel. She is now with the Department of Agricultural Engineering, Purdue University, West Lafayette, IN 41901.

U. M. Peiper and Y. Sarig are with the Agricultural Engineering Institute, Agricultural Research Organization, P.O.B. 6, Bet-Dagan 50250, Israel.

T. Flash is with the Department of Applied Mathematics and Computer Sciences, The Weizmann Institute of Science, Rehovot 76100, Israel.

I. Shmulevich is with the Faculty of Agricultural Engineering, Technion, Israel Institute of Technology, Haifa 32000, Israel.

IEEE Log Number 9040545.

torques. The trajectory-tracking problem deals with controlling in real time the robot’s actual posjtions and velocities to follow the desired values as closely as possible. Many approaches to the solution of these problems have been suggested. In the following, only studies dealing with the trajectory-planning problem will be reviewed. For a robotic system, the minimum-time geometric path of the robot’s end-effector (i.e., the time history of positions) is not necessarily equivalent to the path of minimum Euclidean distance, but depends on the manipulator’s kinematics and its inertial parameters.

Algorithms that are based on a pure kinematic approach to the problem and use various methods, such as linear or nonlinear programming, were suggested by several researchers [lo], [24]. Since these solutions do not consider the dynamics, they are not necessarily optimal. Several algorithms that did consider the dynamics of motion provided a solution to the trajectory-planning problem, but required’ a predefined geometric path as input. Several of these solutions [20], [33] were based on the lineariza- tion of the equations of motions, while others made use of the time-scaling property of robot dynamics. Some of these suggested solutions were based on representing the robot dynamics in Cartesian space [3] while others solved the problem in joint space [32], [34]. To achieve a minimum-time solution, both geometric path and velocity along the path should simultane- ously be determined.

Shin and Mckay [353 developed a method that provides a complete solution to the global optimization problem by solving both subproblems, i.e., trajectory planning and geometric path planning. In their model, the near-minimum-time geometric path was derived while considering both the dynamic and kinematic models of the robot. Their study has shown that the near-minimum-time path of the robot manipulator is a geodesic in the manipulator’s inertia space.

However, most applications require motions through more than two task points. The purpose of this research is to suggest a computationally efficient algorithm that allows determination of the near-minimum-time path between N points, based on Shin and Mckay’s algorithm. When selecting a robot for a given task, the time required by the robot to complete this task is an important factor. Hence, an algorithm that allows us to plan minimum-time trajectories and that can provide a method for testing and comparing the performance of different robots can be quite useful.

The criteria for selecting a robot arm to perform a specific task may include all or some of the following:

1042-296X19 1/0200-0048$01 .00

work space characteristics, including volume, shape and the manipulator’s kinematic redundancy [23], [27], [36],

payload; mobility, i.e., the number of degrees of freedom and the types of independent motions [9]; speeds and accelerations of prescribed motions;

1391, [4Ol;

0 1991 IEEE

EDAN et al.: TASK PLANNING FOR FRUIT-PICKING ROBOTS 49

5) accuracy and repeatability; 6)

7)

structural dynamic characteristics, i.e., stiffness, inertia, damping, and natural frequencies [37]; and economic consjderations: cost, reliability, and maintain- ability.

Since there are a large number of closely interrelated design parameters, optimal robot selection is an extremely difficult and time-consuming task that is not yet automated. Rivin [31] suggested a statistical approach to robot design by determining the optimum configuration according to the probability of occur- rence of loads while evaluating typical cycle performance. In the design phase of a robot, simulation is usually used in order to evaluate robot performance [22], [26], [38].

Time performance of the robot’s task depends on the path the robot follows. Since the minimum-time path depends on the kinematic and dynamic characteristics of the robot, traveling time should be considered when a particular robot is selected for the performance of a given task. The algorithm developed in this work can assist in choosing the most time-efficient kinematic and dynamic structure for a specific task. This algorithm was applied to the specific problem of defining the motions of a citrus-pick- mg robot (CPR), i.e., determining the sequence of fruits to be picked and the path of the robot between consecutive fruit pairs.

Two single-fruit-picking robots have been developed; 1) An apple-picking robot, in CEMAGREF, France [ 121 - [ 161, [29], and 2) a citrus-picking robot (CPR) at the University of Florida. Both systems pick at a rate of one fruit every 4 s. The vision system scans the tree from bottom to top. During this scan, images from the camera are analyzed to detect the fruits. When a fruit is detected, it is immediately picked. This, however, might not be the fastest way to pick the fruits. One possibility to increase the productivity of a robotic system is to minimize the time required for the picking cycle. This can be achieved by preplanning the sequence of robot motions before the beginning of the picking process. The sequence of motions between fruits, as determined by the algorithm presented here, can enable reduction of the total picking time. The algorithm chosen is computationaly efficient. By estimating the time it takes to perform the task for a given kinematic design and for a specified robot’s inertial parameters, the most time-efficient CPR can be chosen.

11. DESCRIPTION OF THE RESEARCH In developing the algorithm for minimizing the task time, the

1) 2)

following assumptions were made:

The locations of the task points are known. No obstacles exist between the task points. Clearly this is an unrealistic assumption. Hence, the model chosen here should enable future extensions of the algorithm that deal with obstacle avoidance. After a task point is reached, the manipulator moves directly to the next task point. The time it takes to perform the task is not considered when deriving the cycle times.

4) The dynamic and kinematic characteristics of the robot are known.

The algorithm combines two stages. The first stage takes into account the robot’s kinematic design and its inertial properties and specifies the near-minimum-time path between all possible pairs of task points. The model used to find the minimum-time path between two points is discussed in Section 111. The second

3)

stage involves the planning of the sequence of motions between the task points in order to achieve global near-minimum cycle time. The problem of passing through N points in minimum time is analogous to the Traveling Salesman Problem (TSP), a well-known problem in operations research. Section IV de- scribes the TSP. The research methods are described in Section V, and the results are described in Section VI. A sensitivity analysis of the chosen path was performed and is described in Section VII. The paper concludes with Section VIII.

111. PLANNING OF MINIMUM TIME TRAJECTORIES The first stage involves the planning of the minimum-time

path between all pairs of task points. The algorithm used here to find the near-minimum-time path between every two task points is based on the model developed by Shin and Mckay 13.51. Therefore, this model is described in detail.

To derive a near-minimum-time path, Shin and Mckay [35] applied Riemannian geometry, which is the study of the metric properties of spaces of an arbitrary number of difhensions. A Riemannian space is described by its metric tensor, which represents the square of the differential line element as a quadratic form in the differentials of the coordinates. In such spaces, distances along curves are calculated by integrating the distance of the differential line element. The geodesic is a curve of minimum distance in Riemannian space. Hence, it is equivalent to a straight line in the Euclidean space. The total distance traveled along a curve in inertia space can be calculated by integrating the following equation:

ds2 = JIJ dq, dq, (1)

where ds is a differential line element and q, and q, are joint coordinates. The geodesics can be derived from the following equation:

d2qj dqj dqk J j , - + [ j k , i ] - - = 0

ds2 ds ds where

4 j is a metric tensor; ds is a differential line element; qi and qi are joint coordinates; and [ j k , i] is the Christoffel symbol of the first kind and is

equal to

1 . 1 dJij dJik dJjk - - + - - - 2 i dqk &j dqj

Shin and Mckay have shown that the total time T to pass along the path between two points is proportional to the geodesic distance S and can be found as follows:

(3)

where S is the distance between the movement end points along the geodesic path length, t is time, and Pm, is the maximum power.

They have proved analytically that the geodesic is the near- minimum-time geometric path under the following constraints:

1) The terms that relate to friction and gravity forces are

2) There is an upper limit on the total power consumed by the

3) The initial and final velocities are zero.

ignored.

robot’s torque motors.

50 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 7, NO. 1 , FEBRUARY 1991

200 I

The conditions under which this optimality applies are not realistic. Hence, Shin and Mckay [35] derived a lower bound on the time required to move the robot between two points and proved that the lower bound increases monotonically with the geodesic distance. Minimizing the lower bound should approximately minimize the traveling time. Another criteria for selecting a near-minimum-time path can be to minimize the product of the path’s length and curvature. In both cases, when the near-optimal path is obtained, the path is found to be a geodesic in the robot’s inertia space. This result was also proven by computer simulations [35]. Therefore, the geodesic, being the curve of the shortest distance between any two points, is the optimal geometric path. In this work, the selection of motion sequence is based on calculating the geodesic distance between all possible pairs of points. Given the above relation between distance and traveling time, calculating the geodesic distance is equivalent to calculating the time.

-

IV. TRAVELING SALESMAN PROBLEM To define the sequence of task points, the TSP algorithm was

applied. This TSP involves routing a salesman through a sequence of known locations (cities) so as to minimize the distance traveled, the time, or any other well-defined function. There are several algorithms that produce an optimal solution to the TSP. These algorithms are based on various optimization techniques such as dynamic programming [2], [4], the assignment problem [4], and integer programming 121.

However, the TSP is one of a number of combinatorial optimization problems for which no efficient algorithms are known. It belongs to a class of problems called NP-complete for which all known algorithms require an exponential time [ 1 11. Consequently, problems involving more than 70- 80 cities can- not be solved in a reasonable time by any of the known methods. To produce near-optimal solutions to the TSP for a large number of points, many heuristic algorithms have been developed [6 ] , [17], [21], [25], [30]; these algorithms have a polynomial time complexity.

In this research, a solution of TSP will define the sequence of motions through the task points in order to achieve minimum time. The application of the TSP to our problem, however, is not straightforward. Some of the problems are as follows:

1) Our minimum criterion is time, but minimum time is not necessarily achieved by following the minimum Euclidean path.

2 ) The TSP is usually a two-dimensional problem. In our case the problem is a three-dimensional one.

Thus, an appropriate cost function that considers these problems needs to be defined prior to the actual TSP application. First, an algorithm to solve the TSP has to be selected. The criteria for choosing such an algorithm are as follows:

1) low computation complexity; 2 ) reasonable storage requirements; and 3) the answer need not necessarily by the true optimum

According to these criteria, several heuristic algorithms, which had a time complexity of O(n2) and could be expanded to the non-Euclidean case, were examined. The tested algorithms in- cluded the nearest-neighbor algorithm (NN), the closest insert algorithm, the minimum spanning tree-vertex penalty method, and the decision tree method. To evaluate the performance of these algorithms, they were first implemented in computer pro-

A A ‘ t One group 8 Groups o f 50 f ru i ts 0 Groups of 30 f ru i ts

NUMBER OF FRUIT ON DIFFERENT TREES

Fig. 1 . The effect of group size on length of tour for random fruit files.

m Groups o f 5 0 f r u i l r

I-

0 ) L , W I 0 140 180 220 260 300 340 380

NUMBER OF FRUIT ON DIFFERENT TREES Fig. 2 . The effect of group size on computation time for random fruit files.

grams, and tests were conducted on various numbers of ran- domly chosen points. Based on the simulation tests carried out and described in detail in [8], the NN algorithm was chosen as the algorithm to be applied in this research because it required the least computing time and converged even for a large number of points, although it did not always result in the shortest path. The path obtained using the NN algorithm relates to the optimal path according to Reingold et al. [30] as follows:

(4)

where N is the path obtained from the NN algorithm, Op is the optimal path, and n is the number of points.

The NN algorithm considers the local optimal condition for choosing the city sequence as follows: Arbitrarily pick one of the cities as the starting point. Among all the cities not yet visited, choose as the next one to visit the one that is closest to the current one. After selecting the TSP algorithm, a feasibility test was conducted in order to confirm that it is possible to apply this algorithm for a large number of points in reasonable computing time. This was tested by implementing the algorithm on three-dimensional sets of collected data points (which repre- sented exact locations of fruits in the tree and will be described in Section V-A. Only the Euclidean distance between the points was considered in these tests. The results of these simulations indicated a time complexity of O(n2) . To further reduce computing time, the effect of dividing the fruits into smaller groups was also examined. Applying the algorithm twice to groups of 50 fruit points resulted in less than half the computing time required to apply the algorithm to a group of 100 points. A comparison of tour lengths resulting from various group sizes is shown in Fig. 1 , while the required computation time on an IBM 3080 is shown in Fig. 2. The total tour length is significantly reduced when applying the algorithm as compared to applying

EDAN et al.: TASK PLANNING FOR FRUIT-PICKING ROBOTS

8 Human picking I/

*/ 3 O c i I I I I I I I I I

DIFFERENT TREES Fig. 3 . Comparison of picking length with human picking and algorithm.

no algorithm at all. This can be seen from Fig. 1: when the algorithm is applied to all the points, the resulting tour length is minimal. However, when applying the algorithm to subgroups of points, the resulting tour increases as the number of points in each subgroup decreases. For groups containing a small number of points, the algorithm is hardly effective.

To further assess the benefits of using the TSP algorithm, the results of the Euclidean TSP, described above, were compared to the Euclidean tour length resulting from the analysis of the human picking strategy. This analysis was based on recordings of the sequence of fruit being picked by a human picker as measured by the measurement instrumentation described in Sec- tion V-A. Since the human picker picks fruits in clusters and tends to advance from one cluster to another, there is no significant difference between the tour length followed by the human picker and the one obtained by the algorithm. This is described in Fig. 3.

However, a machine approaching this task in a systematic way must be programmed in order to decide in what sequence to pick the fruit. The actual TSP cost function applied in this research was the distance traveled along a near-minimum-time path between every two task points. This cost function considers the dynamic and kinematic properties of the robot, and was based on the work by Shin and Mckay [35]. However, the time required to compute the geodesic distance between all possible pairs of task points is very long since it involves the solution of highly coupled nonlinear differential equations. Hence, to obtain results in a reasonable computing time, an estimate of the geodesic distance was derived and was used as the cost function for the TSP. This will be discussed in detail in Section VI-B.

V . METHODS A . Database

To test the proposed algorithm, measurements of fruit locations on 20 Shamuti orange trees were conducted using a three-dimensional ‘ ‘digitizer” with four degrees of freedom. Each degree of freedom was connected to an encoder. A schematic diagram of the “digitizer” is shown in Fig. 4. The data were collected by manually leading the arm tip toward each fruit on the tree and storing the readings of each encoder at this position on a computer. The readings were taken separately for each half a tree. From these readings, the exact location of each fruit was calculated. An example of a typical measured tree is shown in Fig. 5 . These measurements served as the database of the simulation and are fully described by Katz et al. [ 181.

B. Simulation Model The basis of the simulation model is the TSP algorithm. The

cost function considers the robot’s kinematic design and dynam-

51

Plane view

Top view

Fig. 4. Schematic diagram of the measuring device. 1-Stand. 2-Mova- ble platform. 3-Telescopic arm.

0

Fig. 5. A three-dimensional plot of a measured tree.

ics in order to produce the near-minimum-time path. The inputs are the Cartesian coordinates of fruit locations, which have to be transformed into the corresponding robot’s joint coordinates. The cost function was defined as the distance along the geodesic in inertia space. This distance is derived by integrating (2), which requires numerical values for the robot’s inertial parameters such as link masses, moments of inertia, etc. To find the minimum sequence of motions between the fruits, it is sufficient

52 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 7, NO, I , FEBRUARY 1991

CARTESIAN EPUATIONS

TRANSFER TO

COORDINATES EOUATIONS

DIVIDE INTO NORMALITY SUBGROUPS

$ f-&q FUNCTION

CALCULATE

MATRIX

MODEL

/y/*/ SUBGROUP /y/*/ SUBGROUP

Fig. 6 .

to calculate the path distance, which is proportional to the traveling time. The fruits were allocated to subvolumes; for each subvolume, the cost matrix was calculated for the TSP. The cost matrix elements are the estimated geodesic distances between fruit pairs. The output is the sequence of fruits to be picked. The flowchart of the model is presented in Fig. 6.

C. Simulation Tests The simulation tests of the algorithm were performed for a

cylindrical robot. This kind of robot was chosen for the following reasons:

1) Preliminary work on the determination of fruit-bearing zones and distribution in citrus by Katz et al. [18] indicated that the basic motion of a citrus-picking robot should be around the tree and then entering the tree radially.

2) A vertical degree of freedom is needed, since the fruits were clustered at different heights.

3) A cylindrical robot is a basic and common structure in robotics. Dynamic and kinematic parameters for such a robot are known.

The equations of the geodesics for a cylindrical robot are as

Flow chart of the simulation model

follows:

where

ds is the line element differential; J , is the inertia moment around R = 0; K is a constant, present as the mass center of the r joint

does not coincide with the 0 axis when R = 0; R is the length of the radial segment; M , is the R joint mass; M , is the 2 joint mass; dB is the 0 element differential; dr is the r element differential; and dz is the z element differential.

The manipulator size was scaled according to the average tree size and the inertial parameters derived from [35] were accord- ingly scaled. The TSP cost function was determined for a known cylindrical robot by integrating the geodesic distance element as presented by Shin and Mckay [35] as follows:

ds2 = ( J , - KR + M,R2)d02 + M , dr' + M , d z 2 . (8)

At the first stage of this work, the distance along the geodesic path between each pair of fruits was calculated through the following two steps:

1) solving the geodesic equations [(5)-(7)] using shooting methods, which involve guessing the initial direction of the geodesic and refining the initial guess until the curve comes sufficiently close to the desired endpoint [28], and

2) calculating the distance of the geodesic path from the following equation:

i = n

S = dsi i=O i = n

i=O =

where

(( J, - KR + MI R') de; + M , dr? + M , d z f ) (9)

do. = 8 . - 8 .

dr, = ri - I 1 1 - 1

dz . I = 2 . I - z . 1 - 1

and the terms multiplying de, dr , and dz are estimated at the midpoint.

However, since the derivation of the actual geodesic requires solving a set of nonlinear coupled differential equations with two point boundary values, the above steps require computing time in the range of 0.35 s on an IBM 3080 for each pair of fruits. This is unacceptable for real-time applications since it requires approximately 10890 s for an average of 250 points. However, to determine the sequence of the picked fruit, only the geodesic distance is required. The actual minimum-time path between the points can be found after this final sequence has been determined. To reduce computing time, an estimate of the distance of the geodesic path was derived based on the following approxima-

O = M , + (i - M , R ) ( z ) dt'

d2z 0 = Mz-

ds2

where S is the estimated geodesic distance and

D' = ofinal - 'initial (6)

(7)

EDAN et al.: TASK PLANNING FOR FRUIT-PICKING ROBOTS

Fig. 7. Schematic diagram of divising the fruits into subvolumes.

TABLE I

Mode of Travel Subvolumes First Point

V 193,799 V min, H min V 4,6 V min, H max V 2, 8 V max, H max H 192, 3 , 7 , 8 , 9 V min, H min H 4 , 5 , 6 Vmin, Hmax

V, H min/max are the points in each subvolume with the minimum/maximum values of the position values along the corresponding axis.

Hence, in estimating the distance in inertia space, only the joint coordinates of the path's end points and not the actual path were taken into consideration. Theoretically, this estimated distance in inertia space provides an upper boundary on the geodesic distance. To further reduce computation time, the tree was divided into subvolumes, each 1.6 m in height and with an opening angle of 60 degrees, as shown in Fig. 7. The TSP algorithm with the estimated cost function described in (10) was then applied to all points within each subvolume. Two methods of sequencing through the subvolumes were examined:

1) Vertical travel, defined as V , which imposes a travel along the following sequence:

1 + 4 - + 7 - + 8 - + 5 + 2 + 3 + 6 + 9 2) Horizontal travel, defined as H , which imposes the fol-

lowing travel sequence:

1 + 2 - + 3 - + 6 - + 5 + 4 + 7 + 8 + 9

In addition, the TSP algorithm was used to choose the sequence of the subvolumes to be visited. When allocating the points to the different subvolumes, the first points to be visited in each subvolume were defined as given in Table I. The algorithm was programmed in Pascal on an IBM 3080 computer. The solution of the geodesic equations was derived using a procedure from the NAG library.

VI. RESULTS The output motion sequences resulting from the exact and the

estimated TSP cost functions were compared. In 17 of the 25 tested cases an identical sequence resulted from the use of the estimated and exact cost functions. This is described in Fig. 8. The average relative error between the exact distance of the estimated distance was 3.56%. In eight of the 25 investigated cases the output sequence was not identical, but in four of these cases the difference was only in the sequence of two points. In

1200 I - E E

I: 000 I

W W

400 W

0 w

0

0

Fig. 8 . Comparison of exact and estimated solutions of the distance for different trees.

1.0 / I - 2 - 0.0 r I- 0.6

0 t- 0.4

z 0.2

0

W

z

2 a 0 U

5 15 25 35 45 AVERAGE GROUP SIZE

Fig. 9. Computation time versus average group size.

53

geodesic

the remaining cases there was an exchange of two groups of points (for example: if each letter represents a group of points, then one sequence was a-b-c-d-e while the other one was a-d-c-be), but the global sequence was quite similar.

The average computation time of finding the geodesic path between any two points, using numerical shooting methods, was 0.35 s. Consequently, for 40 points in a subvolume determining the exact geodesic distance between all pairs of fruits took approximately 273 s! The average computation time of the algorithm, using the estimated distance, was 0.3 s for an average of 250 fruits. This is a significant reduction in computation time. These indicated computing times are virtual times on an IBM 3080 mainframe system (multitasking, multiuser). To derive meaningful computing times, the program was transferred to an IBM PC AT (12-MHz 80286 CPU). The average computing time of the algorithm to derive the sequence of fruit to be picked, using the estimated cost function when run on an IBM PC AT, is 1.013 s for an average of 250 fruits. The time complexity for applying TSP algorithm for the estimated distance was O(n2) where n represents the average number of fruits in a subvolume (Fig. 9). The computing times are accept- able for real-time applications since they are well within the range of the computing times for the image-processing algorithms that identify the fruits to be picked. Since the output sequence is almost identical when calculated on the basis of either the estimated or exact cost functions, it seems that the estimated cost function should be applied in the first stage of the algorithm because of the significant reduction in computing time.

The distances resulting from both traveling modes V and H for 40 different half-trees, are shown in Fig. 10. For the graphical presentation, the trees were sorted according to the number of fruits. A statistical t-test applied to the results indicated that the distance of the path achieved by the V algorithm was significantly shorter than the one using the H mode. For the vertical way of traveling through the subvolumes, the

54 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 7, NO. I , FEBRUARY 1991

Fig. 10.

c

g7 E - 6 I

I I

Vertical travelling Horizontal travellinp

I I I I I I

DIFFERENT TREES Comparison of geodesic length received by vertical traveling and

horizontal traveling for 40 different half trees.

m TSP travelling 0 Vert ical travelling E

I- o z

- 1 0.9

!I 0.7

0 w 0.5 n v)

0 W W

0 3 DIFFERENT TREES

Fig. 11. Comparison of geodesic length between subvolumes received by TSP traveling and vertical traveling for different trees.

total distance was 5% shorter than the horizontal one. The total distance was reduced an additional 2.5% by applying the algorithm with the estimated distance to determine the sequence of the subvolumes (Fig. 11). This step required an additional computation time of 1%. Despite the fact that the distances between the subvolumes were large, the same sequence was derived when using the TSP based on the estimated function and the exact function in most of the cases. The average relative error between the estimated and exact cost functions for determining the subvolumes sequence was 10%. However, at this step it might be worthwhile to use the exact geodesic distance since the differences between the output sequence, if occurring, have a larger effect on the picking time. Applying the exact function to determine the subvolume sequence takes an average of 10 s of computing time on an IBM 3080.

VII. SENSITIVITY ANALYSIS

A. Dynamic Characteristics of the Robot In the original simulation tests described above, the mass of

the vertical link of the tested robot was assumed to be four times larger than that of the horizontal one. The effect of assuming different inertial parameters for the cylindrical arm were tested by simulating the following cases:

a) MO = 4Mz b) MO = lOM, C) MO = 50Mz

For the first tested case the vertical ( V ) way of traveling was shorter than the horizontal ( H ) traveling mode in all cases. Thus, in this case, in spite of the change in the inertial properties of the robot, the geometrical nature of fruit distribution is still the dominant factor. For case b), the horizontal way of travel

Fig. 12. Different geometric design of simulated trees

was shorter than the vertical way of travel in 50% of the tested cases. In case c), the horizontal way of travel was shorter in 90% of the cases. Additional ratios between the values of the link masses were not tested since they become unrealistic.

B. Fruit Distribution The effect of fruit distribution within a tree was examined by

testing ten trees with random fruit distributions within the tree volume. The fruit locations ( X , Y , Z coordinates) were ran- domized within the volume of the actual tree size. Under these conditions, the resulting V path was found to be shorter than the H path. This indicates that the nature of the fruit distribution within the tree volume does not influence the mode of travel. For a tree with a geometrical design as shown in Fig. 12(a), where its major axis is horizontal, simulation tests indicated that the horizontal way of traveling through the tree is shorter than the vertical one. For a typical citrus tree having a geometrical design as shown in Fig. 12(b), the vertical way was shorter than the horizontal one. This indicates that the spatial geometry of fruit distribution does influence the mode of travel though the tree. Thus horticultural practices (such as tree dimensions) should be considered for selecting the sequence of picking motions of a robotic harvester.

VIII. SUMMARY AND CONCLUSIONS

An algorithm was developed to determine the near-minimum- time path of a robot manipulator, with known dynamic and kinematic parameters, through N given task points. The method was based on calculating the distance of the near-minimum-time path between every two task locations and then defining the sequence of motions using a TSP algorithm. An efficient cost function for the TSP was defined based on the geodesic distance in inertia space.

The algorithm presented here was applied to the specific problem of defining the motions for a citrus-picking robot. To reduce computing time the tree was divided into subvolumes. The TSP algorithm was implemented in each subvolume. Two methods of traveling between the subvolumes were examined. The vertical way of traveling was found to be shorter than the horizontal one. The algorithm was tested for a cylindrical robot on 20 trees. Results of the simulation tests enabled us to assess the influence that the robot's kinematic and dynamic parameters and the fruit distribution characteristics have on the selection of the preferable traveling mode. Crop type and horticultural practices may further influence the selection of a suitable design for the robot. It has a been shown that the geometrical distribution

EDAN et al. : TASK PLANNING FOR FRUIT-PICKING ROBOTS

of the task points should be considered in the design phase of a robot.

Using the TSP based on the estimated geodesic distance produces, in most cases, a sequence identical to the one resulting f rom the exact geodesic distance, but with a significant reduction in computing time. This is an important result for practical applications of the proposed algorithm. After obtaining the task sequence from the algorithm based on the estimated distance between all fruit pairs, the exact path between the points can be found by solving the geodesic equations. Defining the sequence of movements of the robot between subvolumes by using the estimated geodesic distance is adequate. However, in this case, calculating the exact geodesic distance between the subvolumes would result in a significant reduction in the picking time of the robot, without a large overhead of additional computing time.

The algorithm must be further extended in order to deal with obstacle avoidance. In addition, it can serve as a basis for comparing the efficiency and performance of different robot designs. T h e suggested algorithm takes into account the kinematic and dynamic properties of the robot when determining the path of the robot through the task points. Hence, the geodesic distances for various robot configurations can be compared and the robot that gives the minimal geodesic distance can be selected as the most time-efficient one. When choosing a robot for a given task with typical spatial distribution of locations through which the robot must pass, the proposed algorithm can determine the minimum path and assist in selecting the most suitable robot.

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Y. Edan, “Determination of an algorithm defining the motions of a fruit picking robot,” Master’s thesis, Technion, Haifa, Israel, Mar. 1988. A. G. Erdman, T. Thompson, and D. R. Riley, “Type selection of robot and gripper kinematic topology using expert systems,” in J. M. McCarthy, Kinematics of Robot Manipulators. Cambridge: MIT Press, 1987, pp. 202-208. T. Flash and R. B. Potts, “Discrete trajectory planning,” Int. J . Robotics Res., vol. 7, no. 5, pp. 48-57, Oct. 1988. M. R. Carey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP Completeness. San Francisco: W. H. Freeman, 1979. D. A. Grand D’Esnon, Robotic Harvesting of Apples, Agri- mation I , ASAE pub. no. 1-85, 1985, pp. 210-214. D. A. Grand D’Esnon, G . Rabatel, R. Pellenc, M. J. Journeau, and Aldon, “Magali: A self-propelled robot to pick apples,” presented at Summer Meeting Amer. Soc. Agriculture Eng., paper 87-1037, 1987. R. Harrell, Economic, “Analysis of robotic citrus harvesting in Florida,” Trans. ASAE, vol. 30, no. 2, pp. 298-304, 1987. R. C. Harrell, P. D. Adsit, and D. C. Slaughter, Real Time

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Vision Serving of a Robotic Tree Fruit Harvester, ASAE paper no. 85-3550, 1985. R. C. Harrell and P. Levi, “Vision controlled robots for auto- matic harvesting of citrus,” presented at Int. Conf. Agriculture Eng., paper 88-426, 1988. M. Held and R. M. Karp, “The traveling salesman problem and minimum spanning trees,” Oper. Res., vol. 18, pp. 1138-1167, 1970. N. Katz, Y. Edan, Y. Sarig, and U. M. Peiper, “Determination of fruit bearing zones and distribution in citrus,’’ Israel Agre- search, vol. 2, no. 1, pp. 69-86, 1988. B. K. Kim and K. G. Shin, “Minimum time path planning for robot arms and their dynamics,” IEEE Trans. Syst., Man, Cybern., vol. SMC-15, pp. 213-223, Mar./Apr. 1985. A. L. Kornhauser and M. L. Brown, “Optimal trajectories for robot manipulators using state-space networks,” in Proc. IEEE Int. Conf. Robotics Automat., 1985, p. 750. P. Krolak, W. Felts, and G. Marble, “A man machine approach toward solving the travelling salesman problem,” Commun. Ass. Comput. Mach., vol. 14, pp. 327-334, 1971. J. K. Krouse, “Engineering without paper,” High Technol., vol. 6, no. 3, pp. 38-43, 45-46, 1986. A. Kumar and K. J. Waldron, “The dextrous workspace,” presented at Design Eng. Tech. Conf., ASME, paper IO-DET- 108, 1980. C. S. Lin, P. R. Chang, and J. Y. S. Luh, “Formulation and optimization of cubic polynomial trajectories for industrial robots,” IEEE Trans. Automat. Contr., vol. 28, pp. 1066-1073, Dec. 1983. S. Lin and B. W. Kernighan, “An effective heuristic algorithm for the travelling salesman problem,” Oper. Res., vol. 21, no.

B. Novack, “Robotic simulation facilitates assembly line design,” Simulation, vol. 43, no. 6, pp. 298-299, 1985. B. Paden and S. Sastry, “Optimal kinematic design of 6R manipulators,” Int. J . Robotics Res., vol. 7, no. 2, pp. 43-59, Mar./Apr. 1988. W. H. Press, B. P. Flannery, S. A. Tevkolsiky and W. T. Vetterling, Numerical Recipes in C . Cambridge, U.K.: Cam- bridge University Press, 1988. G. Rabatel, “A vision system for Magali, the fruit picking robot,” presented at Int. Conf. Agriculture Eng., paper 88-293, 1988. J. M. Reingold, J. Nievergelt, and N. Deo, Combinatorial Algorithms: Theory and Practice. Englewood Cliffs, NJ: Prentice-Hall, 1983. E. I . Rivin, Mechanical Design of Robots. New York: Mc- Graw-Hill, 1988. G. Sahar and J. M. Hollerbach, “Planning of minimum time trajectories for robot arms,” Int. J . Robotics Res., vol. 5, no. 3, pp. 90-100, Fall 1986. V. Scheinman and B. Roth, “On the optimal selection and placement of robotic manipulators,” in Theory and Practice of Robots and Manipulators, A. Morecki, G. Bianchi, and K. Kedzior Eds. K. G. Shin and N. D. Mckay, “Minimum time control of robotic manipulator with geometric path constraints,” IEEE Trans. Automat. Contr., vol. 30, no. 6, pp. 532-541, June 1985. -, “Selection of near minimum time geometric paths for robotic manipulators,” IEEE Trans. Automat. Contr., vol. 31, no. 6, pp. 501-511, June 1986. R. Vijaykumar, K. J. Waldron, and M. J. Tsai, “Geometric optimization of serial chain manipulator structures for working volume and dexterity,” in Kinematics of robot manipulators, J. M. McCarthy, Ed. Cambridge, MA: MIT Press, 1987, pp.

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Yael Edan received the B.Sc. degree in computer engineering in 1984 and the M.Sc. degree in agricultural engineering in 1988, both from the Technion, Israel Institute of Technology, Haifa, Israel She is currently workmg toward the Ph.D. degree at Purdue University, West Lafayette, IN, where her research concerns the development of a robotic melon harvester.

Between 1984 and 1988, she worked as a Computer and Research Engineer at the Institute of Agricultural Engineering, ARO, Bet-Dagan,

Israel, where she developed hardware and software for a variety of computer applications in agriculture.

Tamar Flash received the B.S. degree in 1972 and the M.S. degree in 1976 from Tel-Aviv University, Israel, both in physics. She received the Ph.D. degree in medical physics in 1983 from the Massachusetts Institute of Institute. Cambridge, where she was enrolled in the Med- ical Physics Medical Engineering Program at the Harvard-MIT Division of Health Sciences and Technology.

From 1983 to 1985, she completed two years of postdoctoral research at the Massachusetts

Institute of Technology’s Department of Brain and Cognitive Science and the Artificial Intelligence Laboratory. In 1985, she joined the Department of Applied Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel, where she is currently a Senior Scientist. Her research interests include computational neuroscience: motor control, robotics, and artificial intelligence.

Uri M. Peiper received both the B.Sc. and M.Sc. degrees from the Technion, Israel Insti- tute of Technology, Haifa, Israel.

Since 1962 he has been working at the Insti- tute of Agricultural Engineering, ARO, Bet- Dagan, Israel. His work there began with research into hillside farming, cultivating of stony and sloped lands, developing harvesting machinery for vegetables for the frozen food indus- try where quality and absence of bruises is of prime importance, and the development of ma-

chinery for the cultivation and harvesting of peanuts. In 1972, he was nominated as director of the department for testing and instrumentation,

which is the Ministry of Agriculture’s official testing laboratory for Agricultural Machinery. He developed different apparatus, testing pro- cedures, and transducers for field machinery, greenhouses, controlled- climate animal houses, and other agricultural equipment testing and research. His main interest at present is in introducing computers and electronics to Israel’s agriculture, in developing special transducers and expert system software for dairy farming, and in developing a robot for melon harvesting.

Mr. Peiper is member of the Central Committee for Machinery Standards at the National Institute for Standards, and was President of the Israeli branch of the Instrument Society of America from 1983 to 1985.

Itzhak Shmulevich received the B.Sc. (Hons.) degree in 1971 from the Technion, Israel Insti- tute of Technology, Haifa, Israel, and the M.Sc. and Ph.D. degrees from the Technion 1985 and 1980, respectively, all in agricultural engineering.

From 1981 to 1983, he was visiting Associate Professor at, Texas A&M University Depart- ment of Agricultural Engineering. Since 1983, he has been on the Faculty of the Department of Agricultural Engineering at the Technion. He is

now a Senior Lecturer and on sabbatical leave with Texas A&M University. His research interests have been in the areas of the physical properties of agricultural materials and soil machine/structure interac- tion in tillage and traction.

Yoav Sarig received the B.Sc. and M.Sc. degrees from the Technion, Israel Institute of Technology, Haifa, and the Ph.D. degree from Michigan State University, East Lansing, MI, all in agricultural engineering.

He is currently the leader of a research group at the Institute of Agricultural Engineering, ARO, Bet-Dagan, Israel. He has held the positions of Director of the Institute of Agricultural Engineering and Deputy Director of the ARO. His areas of interest are the mechanical harvest-

ing of fruits and vegetables, the physical properties of agricultural products, and nondestructive postharvesting technologies for fruits and vegetables.

Dr Sarig is a member of the ASAE, the IAAE, and the International Society of Citriculture.

near-minimum-time task planning for fruit-picking robots

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