4. Simulation results
The risk-reward optimization problem (8) for human-robot collaboration is analyzed for robotic citrus harvesting in Florida, USA. However, we emphasize that this framework is not restrictive to the presented application, and it can be suitably modified for use in other agricultural operations requiring human-robot collaboration.
4.1. Experimental setup
The developed framework is validated in a simulated grove environment using citrus production data for 2016–17 obtained from the United States Department of Agriculture (USDA), National Agricultural Statistics Service, Southern Region, Florida Field Office. Relevant to the problem in hand, data shown in Table 1 was obtained for Valencia, Navel, Early-midseason, white grapefruit, and red grapefruit. The results presented in this section consider simulated citrus groves of 100 acres for fresh fruit production for the above fruit varieties. Based on the USDA statistics, we obtain production parameters for the simulated groves as shown in Table 2. The production data in Table 2 can be directly obtained by scaling down the data in Table 1 for the simulated groves of 100 acres. The remainder of this section provides values of the parameters used in model (8) and discusses the experimental setup for the presented human-robot collaboration framework.
During the harvesting operation, any loss in efficiency at early stages of the operation can result in higher economic losses because the number of fruit that may not get harvested will be higher if the system continues to operate at reduced efficiency. Subsequently, human collaboration can be economically beneficial at early stages. However, at later stages of the operation, i.e., when the number of fruit remained to be harvested is lower, it is not certain whether human collaboration will always be economically beneficial. Specifically, the labor cost associated with human collaboration may offset any economical advantage gained through improved production. Therefore, to gain insights into the question of when should a human collaborator interact with a robotic system that will be economically beneficial to the co-robotic operation, different stages of the harvesting operation are considered. The stages correspond to the percentage of fruit remaining to be harvested, namely, we consider stages with 100%, 75%, 50%, 25% and 15% fruit remaining to be harvested. The parameter ci, i.e., the cost of reduced production due to unit loss in efficiency of i, in (6) and (8) can be obtained from the value of production in Table 2 for each fruit variety. The values of ci corresponding to different stages of harvesting for the considered fruit varieties are shown in Table 3. The presented results assume that ci is constant across all the components, i.e., ci = cj , ∀i =∕ j and i, j ∈ I . In other words, the loss in efficiency of each component contributes equally to reduction in the value of production. This is a mild assumption since the loss function in (6) is designed to accommodate differences in ci. Also, it is to be noted that ci is directly proportional to the value of production and thus the orchard size; the cost of reduced production per unit loss in efficiency is higher for larger orchards.
The total service time in (8f) was chosen to be T = 20 minutes, and the prescribed return λ in (8d) was selected to be 100 dollars, which accounts for the labor cost expended in servicing the harvester for time T. From (8d), since λ is a lower bound on the expected return, human collaboration is economically beneficial only when the expected reward, i.e., the increase in the value of production after human collaboration, is greater than or equal to 100 dollars. Net profit is obtained as the difference between the increase in the value of production and the cost of labor for time T, where the labor rate was considered to be 30 dollars per hour. The parameter λ may also be chosen to include economic losses as a result of downtime in other operations when the human collaborator services the harvester. In general, larger values of λ can be selected to achieve higher expected reward, which, however, may reduce human involvement if the increase in the value of production is less than the prescribed λ.
The confidence level α in the objective function of (8) was selected to be 0.9. As described earlier, the risk measure HMCR in (4) is considered with p = {1,2,3}. For the selected value of α, the case of p = 1 relates to reducing the expected losses in 10% worst-case operating scenarios, while the loss is calculated via higher moments when p = 2, 3.
The robotic harvesting operation is considered to be comprised of the following seven components: Fruit detection, fruit localization, 3D mapping, robot path planning, visual servo control, fruit detachment, and fruit storage, which constitute set I. In practice, as stated in Assumption 1, the probability distribution of the nominal efficiency ηi, i = 1, ⋯, 7, of these components can be obtained for a given robotic harvester and a fruit variety using the harvester's historical performance data. However, as the robotic citrus harvesting systems are still being investigated and no commercial solutions are available, such probability distribution functions of component efficiencies cannot be found in literature. Therefore, the presented results employ the best estimates of the component efficiencies based on the data available in the current state of knowledge and the authors' experience to make the model as realistic as possible. Fig. 2 shows the assumed probability distributions of the nominal efficiencies ηi,∀i = 1,⋯,7, of the components modeled as lognormal distributions, which are commonly used in reliability analysis to model uncertainties in the system behavior. In (8), N = 10000 operating scenarios are generated by randomly drawing samples ηik,k = 1,⋯ ,N, from the distributions of ηi,∀i. For simplicity, the same N scenarios are used for the different fruit varieties under consideration. It is to be noted that the mathematical formulation presented in the paper does not make assumptions with respect to the distribution functions of ηi,∀i. In other words, any arbitrary distributions of these random variables as observed in practice can be accommodated without any change in the model.
We performed 10 experiments with different values of the component efficiencies ̂ηi,i = 1,⋯,7, as shown in Table 4 to determine whether human collaboration is economically beneficial in the given experiment, and, if so, what is the optimal servicing policy in terms of servicing times for each component. These experiments can be viewed to simulate different operating points of the system, when the components are operating at various efficiency levels. For each experiment, the component efficiencies are obtained via simple random sampling from the corresponding probability distributions given in Fig. 2, where the drawn samples ̂ηi,i = 1,⋯,7, represent an operating point of the system. Ten such operating points are obtained to yield experimental configurations given in Table 4, such that each row represents an experiment with the associated simulated efficiencies of the components.
Section 4.2 describes the results of the risk-averse collaboration model presented in (8) for the 10 simulated experiments for each of the five harvesting stages (corresponding to remaining harvest of 100%, 75%, 50%, 25%, 15%) and p = {1, 2, 3}. The model in (8) is solved for each configuration of the triplet {experiment, harvesting stage, p}. For any instance of model (8) solved, the parameter values used (8) are given by: ci corresponding to the harvesting stage and fruit type pro- vided in Table 3; the corresponding experiment row in Table 4; the N = 10000 randomly drawn operating scenario values for η from the probability distributions in Fig. 2; and values for T, p, λ, α provided in the preceding discussion. The solution algorithm was implemented in C++, and CPLEX 12.6 mixed-integer programming solver was utilized to solve problem (8). The computation hardware included an Intel Xeon 3.50 GHz PC with 64 GB of RAM, and a Windows 7 64-bit environment. The solution of (8) is obtained for each experiment in terms of the optimal service time xi and the optimal objective value (i.e., the risk of economic loss). Also, the expected reward is given by the value of the left-hand-side of constraint (8d).
4.2. Results
For brevity, we present in-depth results and discussions for the Valencia orange while only the key results pertaining to economic benefits of human-robot collaboration are shown in Fig. 8(a)-(d) for other citrus varieties. It should be noted that the results and discussions for these citrus varieties are analogous to that of the Valencia orange.
Fig. 3 shows the main result of the developed optimal collaboration framework for harvesting Valencia oranges for different experiments when p = 1 and the remaining harvest is 100%. The bar graphs in Fig. 3 show the efficiency of the components before and after human collaboration along with the obtained optimal service time for each component. Component efficiencies before human collaboration are shown as gray bars, which correspond to the values in Table 4. As discussed in Section 3, the improvement in efficiency after servicing component i for time xi is ̂ηi γik xi . Since γik in (7) is computed over N operating scenarios ηik, we consider the median increase in ̂ηiγikxi as a measure of improvement in efficiency. In Fig. 3, the median improvement in efficiency, median(̂ηiγikxi), of each component is shown as a green bar stacked on top of the measured component efficiency (gray bar), thereby the aggregate (gray + green bars) visualizes the component efficiencies after human collaboration. The blue error bars show the least and the largest efficiency of the components in N scenarios after human collaboration. It is noteworthy that deterministic values for improvement in efficiency of individual components are intractable to obtain, since the model involves finding optimal decision over stochastic component efficiencies, and the objective function involves minimizing the global economic loss over all the components. Consequently, the results provide median improvement in efficiency of each component, and the error bars indicate the deviation in the aggregate efficiency over N random scenarios after human collaboration.
The function γik (̂ηi , ηik ) in (7) is a measure of serviceability of component i in scenario k. As such, γik provides insights into which components to service in the limited time T to realize the maximum economic benefit. Fig. 4 shows the average serviceability, E(γik(̂ηi,ηik)) ∈ [0, 1), of component i over N scenarios for the case when p = 1 and the remaining harvest is 100%. By design, E(γik (̂ηi , ηik )) = 0 corresponds to no serviceability and the serviceability improves as E(γik (̂ηi , ηik )) approaches 1. This can be verified by considering the expression for the median improvement in efficiency median(̂ηiγikxi). A given improvement in efficiency can be obtained with less service time xi when E(γik(̂ηi, ηik)) is large (i.e., the component is easy to service). For a component with less serviceability, i.e., smaller E(γik(̂ηi,ηik)), the service time would be higher to achieve the same improvement in efficiency. The limiting case corresponds to E(γik(̂ηi, ηik)) = 0 (or γik = 0, due to positive semi-definiteness), when no improvement in efficiency can be realized for any arbitrarily large service time.
Fig. 5 shows the matrix plot of when human collaboration was sought in the 10 experiments during different stages of the harvesting operation. The value of 1 indicates that human collaboration was economically beneficial and sought; 0 indicates that human collaboration was not sought. The matrix plot is shown for different values of p = {1, 2, 3} in Fig. 5. This plot provides a graphical answer to when should a human collaborator interact with a robotic system that will be economically beneficial to the co-robotic operation. It can be seen that the need for human collaboration varies with the harvest remaining as well as the experimental scenario (i.e., component efficiencies).
The risk of economic loss (dollars) versus the expected reward (dollars) is shown in Fig. 6 for p = {1, 2, 3}. Each line segment corresponds to an experiment and the data points on each line correspond to the harvesting stages.
Fig. 7 shows the net profit (dollars) after human collaboration in each experiment for various harvesting stages and the case when p = 1. Fig. 8 shows the net profit (dollars) for Navel, Early-midseason, white grapefruit, and red grapefruit, respectively.