[01] Decision and Control of human-centered robots
Jaemin Lee / Minkyu Kim
Topic: Modelling trust of robot to human operator while robot's navigation with human
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(1) Introduction
Social Cognitive Theory (SCT) is able to describe individual behavior in the process of some activities. Here, we tried to figure out the effect of trust between human and robot in the process of navigation. In our real life, lots of environments and pedestrians are surrounded, thus, the robots' locomotion including navigation is very difficult. For this reason, we should guarantee safety for human being as long as the robot is operating. As a solution for this safety guaranteed navigation, the trust model between human and robot could be applied to the system. Trust-based behavior can improve both safety and efficiency of robot navigation. In this work, based on the principle of SCT, we modified the model for describing our own problem which is to maximize trust between human and robot. Especially, we focused on building a simple model to show the detail how to improve the trust between human and robot in navigation task. Then, the model predictive controller is applied to the proposed model, and then, we show how to improve the trust between human and robot for making more perfect task execution.
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(2) Problem Definition
For safe navigation of robots with human operator, many factors should be considered such as total length of path, obstacles, average velocity of robot, and so on. There exist a lot of complicated factors that can affect the completeness and efficiency in the navigation task. Our philosophy is that the trust between human and robot can influence the performance of robots. In addition, robot's behavior and human's guidance can construct the trust between robot and human. If there is no experience that robot and human are collaborating, it is difficult to define and measure the trust between human and robot. Therefore, we specified the relationship between a robot operator and a robot and then it is assumed that the robot operator guides the robot several times and some obstacles including pedestrians arbitrary appear in each trial.
Fig. 1. Illustration of the problem in navigation: Can a robot trust him/her or not?
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(3) Models using Fluid Analogy
The dynamic system behavior is described by the similar way to social cognitive theory (SCT) model. We induced the fluid analogy for the SCT model. The model consists of six tanks and three inputs. The detail informations are listed as followings:
- Tanks
- Experiences (η1)
- Collision Rate (η2)
- Speed of Robot (η3)
- Success Rate (η4)
- Operating Time (η5)
- Trust of Robot (η6)
- Inputs
- The number of experience (ξ1)
- Obstacles and Pedestrians Collision (ξ2)
- Speed of Robot (ξ3)
As a first step, we defined the trust of robot influenced by success rate of the navigation task and operating time for the navigation. The trust of robot is simplified by these two terms. The reason why the trust of robot is defined by theses two terms is that task completeness and short operating time are primary goals in robot navigation. Here, we introduce undesired two cases in which the trust of robot becomes low:
- High success rate, but very long operating time
- Very fast navigation, but low success rate
In addition, we assumed that the experience and collision rate are correlated to success rate. If the robot has a lot of experience with human operator, which means that robot has more information, success rate would increase. However, the success rate may decrease due to the collision rate, which means collision rate has negative effect on success rate. In case of operating time, we set three contributors which are experience, collision rate and speed of robot. Many experiences and high speed can contribute to reduce operating time, but, collision rate may increase operating time. Moreover, the high speed of robot can cause to increase the risk of collision. Fig. 2. shows the model including the correlations.
Fig. 2. Modelling trust-based navigation using fluid analogy
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(4) Mathematical Description
The model shown in Fig. 2. can be described as mathematical forms. Each tank is formulated as 1st order differential equation and the time delay and disturbances are not considered in this work.
The equation can be rewritten as following form:
- input: u = [ u1, u2, u3 ] = [ ξ1, ξ2, ξ3 ]
- state: x = [ η1, η2, η3, η4, η5, η6 ]
- output: y = [ η1, η2, η3, η4, η5, η6 ]
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(5) Validation of System Model
In this section, we validate the proposed model in order to confirm our logics in the modelling. The main philosophy of the model is that experience can affect the trust of robot and then the actual speed of the robot is regulated by the trust. To obviously show this, two different type of simulation results are compared. One of them is the result of robot navigation in terms of relatively low experience and the other is that with high experience. Specific variables and coefficients are described as below:
- Variables
β = [ β62, β32, β63, β14, β24, β15, β25, β35, β46, β56 ] = [ -0.1, 0.1, 0.3, 0.4, -0.35, -0.15, 0.1, -0.1, 0.56, 0.2 ]
γ = [ γ11, γ22, γ33 ] = [ 1, 1, 1 ]
τ = [ τ1, τ2, τ3, τ4, τ5, τ6 ] = [1, 1, 1, 2, 1, 3 ]
- case 1 : Low experience (10) with pulse type collision rate (5) and constant input speed of robot (2)
- case 2 : High experience (20) with the same condition
a. case 1 (Low Experience)
In this simulation, we focus on figuring out the relationship between experience, the trust of robot and actual speed of robot. Fig. 3 shows the designed inputs (experience, collision rate, desired speed of robot). In Fig. 4, there are four graphs indicating each state of system. Because of the effect of the experience, the trust of robot increase (about 1.9), even though there are perturbation which comes from collision rate. Based on the increased trust of robot, actual speed of robot also becomes greater (about 2.5) than the initial desired velocity input (2.0). It means that the robot automatically regulate own speed based on the trust to human operator.
Fig. 3. Input condition ( ξ1, ξ2, ξ3) for low experience: Case 1
Fig. 4. States of system in terms of the given inputs (η1, η2, η3, η4, η5, η6): Case 1
b. case 2 (Hight Experience)
The other simulation shows the effect on the higher experience than the previous case. The collision rate and desired initial speed of robot are designed as the same manner of the previous case 1. However, we modify the experience input as higher value (20) than that of case 1. The inputs of the case 2 is presented in Fig. 5. The simulation results provide us that higher experience cause higher trust of robot, and then, robot is able to increase its speed in the navigation task. Although the simulation results are very simple, it is very straightforward.
Fig. 5. Input condition ( ξ1, ξ2, ξ3) for high experience: Case 2
Fig. 6. Fig. 4. States of system in terms of the given inputs (η1, η2, η3, η4, η5, η6): Case 2
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(6) Model Predictive Control for The System
Extended from the last section, we can implement Model Predictive controller(MPC) to minimize pre-defined cost function. The control strategy for intervention design must incorporate the defined requirements and constraints for navigation robot. As a first step, we demonstrated a simple example that the controller minimize cost function which is error of trust state (η6-ηdes), in the below equation, which means the first term.
- Example 1 (using Python)
This example shows that how many experiences and how much speed should be regulated in order to maintain desired trust level of robot. Therefore, we designed two inputs, which are experience and speed of robot, and the other input (collision rate) is defined as constant value. Fig. 7. shows the input data which comes from MPC. The experience increases and the speed of the robot is also regulated for guaranteeing the trust of robot as the desired value. 
Fig. 7 Input variables designed by Model Predictive Controller
Fig. 8 shows the states controlled by the proposed controller. Although these results are very simple, they are straightforward in the logics. The trust of robot (η6) becomes the desired value (10). The operating time reduction and the success rate are also saturated, it means that we figure out the control command to maintain the trust of robot under the constant collision rate.
Fig. 8. States of the system controlled by Model Predictive Controller
- Example 2 (using Matlab Simulink )
The aim of the control design will be directed not only to follow a reference of trust between human and robot (to maintain navigation performance) but also to minimize inputs. A standard quadratic cost function is used to calculate the decision vector for the optimization problem as
The main goal is to achieve the required trust. The reference output set point (measured outputs) is y_r =[y_6] , where y_6 is trust and. Proposed MPC algorithm is designed to obtain desired inputs for ur=[u_r1 u_r2], where
u_r1 is the experience of navigation and u_r2 is the velocity of robots. Another inputs for plant, or collision rate is not included in this manipulated variable because collision rate is determined by environment. This concept is well fit to the situation that human and mobile robot do navigation task together with trust model. So overall block diagram of MPC controller and plants are described as shown.
Fig. 9. plant model in Simulink
Fig. 10. MPC controller & plant model in Simulink>
<Scenario 1> Collision rate is simple step input (= structured environment )
<simulation settings>
p=5, n=3, u_min=[0 0], u_max=[200, 30], du_max=[3, 15], du_min=[-3,-15], ymax=1000, w_u=[0.05 0.05], Q_y=3;
Fig. 11. Sates controlled by MPC (Scenario 1)
scenario 1 stands for structured environment, desired Trust was given with step function and our system show that input and output make system converge with its reference.
<Scenario 2> Collision rate is rapidly changing (= unstructured environment )
Fig. 12. States controlled by MPC (Scenario 2)
Scenario 2 is little different from scenario1 in that collsion rate is changing. It means that environment is dynamic or unstructured. In this situation experience has to slightly increase to maintain desired trust value in higher collision rate.
In fact, two input sources can be regarded as human factor(Experience) and robot factor (velocity). From the perspective of shared control, to find each degree of role factor according to surroundings can be quite interesting research issue in navigation field. So this MPC approach might figure out two inputs to maximize trust in navigation. More complex model can be simulated with detail input and output constraints.
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(7) Conclusion
We tried to implement to model and control apporaches for trust model when human and robot do navigation task together. For simplicity, we assumed that trust model is defined by task success ratio and operating time. We validate our plant model with first order differential equations and insert Model Predictive Control algorithm to follow desired value with optimal input.
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References
[1] C. A. Martín et al., "A Dynamical Systems Model of Social Cognitive Theory," 2014 American Control Conference, Portland, OR, 2014, pp. 2407-2412.
[2] C. A. Martín et al., "A Decision Framework for an Adaptive Behavioral Intervention for Physical Activity Using Hybrid Model Predictive Control," 2016 American Control Conference, Boston, MA, 2016, pp. 2576-3581
Source Codes (Python and Matlab) : https://bitbucket.org/JaeminLee87/me-396d_robotics/src/007ebd86176a301989c7ad5df829f52bc79cb33d?at=master