Swarm Body: Embodied Swarm Robots

The sense of embodiment is a fundamental product of the human mind. A key aspect of embodiment is the sense of body ownership. Body ownership is “the sense that I am the one who is undergoing an experience”  [12]. Botvinick and Cohen [5] and others revealed that humans can perceive the sense of body ownership toward artificial objects with synchronous visuotactile stimulation. In addition to synchronous visuotactile stimulation, synchronous visuomotor feedback also affects embodiment [57]. Synchronous visuomotor feedback sheds light on another key aspect of embodiment: the sense of agency. The sense of agency is “the sense that I am the one who is causing or generating an action” [12]. From the perspective of the sense of body ownership and agency, researchers have expanded their knowledge of various aspects of human embodiment, such as its temporal [59, 62, 63, 72] and spatial [18, 22, 41, 53] characteristics, as well as the embodiment possibilities of artificial bodies.

With advancements in VR technology, many scholars have explored the embodiment possibilities of various non-biological bodies, including elongated [31], invisible [28], discontinued [50, 70], scrambled [27], re-associated [29, 30], shared [11, 13, 69], multiple [44], and supernumerary [2, 20] body parts, as well as noncorporeal objects [42], through synchronous visuomotor feedback using VR. These studies revealed that humans can embody not only their innate body parts with their original configurations but also those augmented or modified in certain ways as well as non-corporeal objects. This knowledge can be applied in robotics to develop embodied robots that realize intuitive interactions, similar to human interactions with innate bodies.

Robot embodiment has been studied for on-site and remote intuitive robot manipulations to interact with the environment physically. For example, Aymerich-Franch et al.  examined whether non-human–looking humanoid robot arms could be perceived as the user’s arms and reported that the levels of embodiment were similar between human- and non-human–looking arms [4]. Takada et al.  showed that a single user could feel a sense of agency toward multiple robot arms while playing ping-pong with two opponents [68]. Most studies on virtual and robot embodiments have focused on body configuration similar to the human biological body. Recently, researchers have begun to explore the embodiment of Supernumerary Robotic Appendages (SRAs) or unconventional body parts, such as additional limbs [58, 71, 76], hands [24, 77], fingers [51, 61], joints [36], and tails [20]. The embodiment of an SRA or an unconventional body has the potential to augment human abilities and interactions with the environment, which are currently limited by the innate body configuration. For example, tentacle limbs offer greater freedom than innate human limbs. Similarly, swarm robots have the potential to provide robustness, flexibility, and scalability to a human body that conventional humans do not possess. Therefore, this study is the first to investigate the embodiment of a swarm to realize embodied swarm robots.

Swarm robots operate such that many small robots of similar types cooperate with each other to exert significant effects. This unique property endows the devices with robustness, flexibility, and scalability. In HCI, actuated objects were used as tangible user interfaces (TUIs) to allow users to interact with the digital world through physical tabletop objects [10, 48, 49, 52, 54, 65]. Zooids [34] expanded these tangible interaction possibilities by introducing tabletop swarm robots and introduced the concept of Swarm User Interfaces (Swarm UIs). This design space has been further extended by developing new robots [8, 35], adding new action possibilities to robots [40, 45, 66, 67, 78, 79], building mixed reality system cooperative with projected images [15, 16], and introducing interfaces that seamlessly connect the digital and physical worlds [19, 21, 38, 46]. Fundamental applications include sensing, visual feedback, haptic feedback, shape presentation, object representation, object actuation, environmental adaptation, and collaborative actions. As such, the multi-agent nature of swarm robots has realized a flexible and scalable interface between the physical and digital worlds.

This multi-agent nature makes user control of swarm robots challenging. Three main approaches have been adopted to control swarm robots: predefined, physical, and synchronized. Many previous works have used predefined control in which a certain action or state triggers a certain robot’s program. Kim et al.  compared various human-control modalities and strategies for predefined swarm robot control and provided guidelines [25]. Another approach is to apply physical laws to robots so that users can expect and control their behavior [21, 38]. Synchronized control, where swarm robots act in sync with corresponding remote or virtual objects, is also widely used, particularly in the context of TUIs, VR, and physical telepresence. For example, HoloBots [19] move such that their positions match the positions of the corresponding robots or the user’s body part in the remote physical world. These methods enabled physical interactions with the digital or remote physical worlds. However, these swarm robots do not aim for embodiment or seek to represent human body parts. By embodying swarm robots, the robot user can interact with the environment more intuitively and precisely while maintaining the power of the swarm robots. In addition, the observer can understand the intentions of robot users more intuitively and precisely while feeling more humanness or affection toward the robots. In human-robot interactions, an early attempt to control swarms in an embodied manner using fingers has already been made [23]; however, the robots do not represent the hand well, and their embodiment and interaction opportunities have not been investigated. Inspired by these, this study explores the interaction opportunities of embodied swarm robots.

Step (1) sets the subgoal formation for the swarm robots. Ideally, these positions are where the robots are; hence, this step affects the visual representation of the hand as well as the usability and interaction opportunity.

Because this step is similar to the virtual body representation, we took inspiration from that research. We explored abstract virtual body representation approaches that are applicable to two-dimensional representation because we applied them to tabletop swarm robots. Consequently, we identified two main approaches: point- and silhouette-based [74]. The point-based approach represents the body with a number of points corresponding to certain points on the body, whereas the silhouette-based approach represents a body with a solid shape in a single color. In other words, the former focuses on specific points, and the latter focuses on overall shapes. Based on these approaches, we designed two algorithms to obtain a set of subgoal positions from hand position, orientation, and shape, shown in Figure 3.

Once the subgoal positions are obtained, they are assigned to the robots. Two assignment methods are considered: static and dynamic. The static method assigns a specific subgoal position to the same robot. In dynamic assignment, the subgoal positions are constantly reassigned to realize smoother transitions from one hand shape to another. This problem is defined as an assignment with variable subgoal formation. This was attributed to the linear sum assignment problem [7]. This problem can be solved by using the Hungarian algorithm [32].

The bone-static algorithm will offer more predictable robot movements than the other algorithm because the same robots always follow the same parts of the hand owing to the fixed subgoal formation and assignment. By contrast, the dynamic assignment can avoid potential collisions of the robots. For example, when the user turns the hand from facing upward to facing downward, dynamic assignment reassigns robots so that they do not have to flip their positions.

When the subgoal positions are constantly generated, and the current subgoal positions cannot be mapped to the past ones (i.e., when the silhouette-based subgoal formation is used), the static assignment cannot be applied because the same subgoal position does not exist at the next instant. Therefore, the possible subgoal formation generation and assignment algorithms are: bone-static, bone-dynamic, and silhouette-dynamic.

Once pairs of a robot and subgoal position are determined, path planning and following robot control are executed. To plan paths and move multiple robots to their assigned subgoal positions, the Reciprocal Velocity Obstacles (RVO) algorithm [73] was used. The RVO algorithm is an extension of the Velocity Obstacle concept, which offers navigation among passively moving objects by treating them as obstacles in the velocity space. The RVO algorithm incorporates the assumption that other actively moving objects perform a similar collision avoidance behavior to the Velocity Obstacle and realized navigation among both passively and actively moving objects. More precisely, the RVO algorithm with nonholonomic constraints [64] was used as most of the swarm robots, including the tabletop swarm robots we use, are nonholonomic.

The experiment involved 10 participants (6 males, 4 females; average age: 24.20 ± 2.57 SD). Participants were sourced from a recruitment post on social media. All participants were right-handed with normal or corrected vision and were unaware of the experiment’s purpose. Half of the participants had minimal VR experience, while the other half had extensive experience. Participants signed a consent form regarding the experiment and were compensated with approximately $16 in Amazon gift cards. The ethics review board approved the experiment.

The experiment program, implemented using Unity, simulates and visualizes swarm robots and runs on a Windows-based computer. A Meta Quest 2 HMD¹ tracked participants’ hands and provided visual feedback based on the Unity visualization. Participants wore noise-canceling headphones playing white noise to block external sounds. An iPad collected the post-trial questionnaire responses. In the VR, robots appeared on a table in front of the user as cylindrical bodies, a common form in HCI research [34]. The subgoal formations are located on the table surface, and they are not translated in the horizontal direction from the participant’s actual hand; i.e., there was no spatial discrepancy in the horizontal direction between the participant’s hand and the robot’s subgoal formation.

We examined parameters (independent variables) specific to swarm robots. The potential design parameters include robot’s latency, speed, acceleration, size, color, shape, density, and control algorithm. As it is not feasible to examine all the parameters, we focused on some of the parameters that are more unique for swarm robots: size, density, and control algorithm. We excluded latency from the examined parameters because the effect of delay on embodiment is not unique to swarm robots and has been investigated in visual-motor synchronicity [62]. In a system with program-robot communication, we should consider the effect on embodiment owing to the sum of the delay caused by the robot’s movement performance and the delay caused by this communication. The first delay was excluded from this experiment by setting the robot’s wheel speed to 400 mm/s with which we did not see much latency in the pilot study. The second delay was excluded by removing the latency in the program-robot communication.

We used a factorial design with three factors: 2 levels of the robot’s size, 3 levels of the density, and 3 levels of the control algorithm. The independent variables examined were robot’s size (30 mm and 20  mm), density (sparse, medium, and dense), and subgoal position generation and assignment algorithm (bone-static, bone-dynamic, and silhouette-dynamic). All variables were within participants.

The experiment was conducted in the virtual environment shown in Figure 4. Participants were tasked with guiding swarm robots to a target sheet while making specific hand signs using their right hand. The targets were green hand-shaped sheets with variations shown in Figure 5. Participants were instructed to change their hand signs to the target shapes when reaching the targets. A variety of hand signs were provided so that participants could explore the pros and cons of the subgoal position generation and assignment algorithms in various situations. For example, the silhouette-based algorithm may have fewer collisions than the bone-based algorithm for the rock and scissors hand signs, but they may not make a big difference for the paper hand sign. In the preliminary testing, the static and dynamic subgoal assignment algorithms resulted in very different robot behaviors (i.e., the static assignment sometimes caused the robots to get stuck, while the dynamic assignment did not) when the hand was flipped over. Therefore, the reversed paper hand sign was also provided.

Participants were instructed to move the swarm robots representing their right hand to the purple starting area at the beginning of each task by moving their right hand. Once all the robots have stayed in the starting area for two seconds, a green hand-shaped reaching target appeared either at the left-front or at the right-front of the starting area. Specifically, the target appeared 300 mm to the rear and 173 mm to the side from the center of the starting area. The participants were instructed to reach for the object with the hand sign and to fit the robots in the target area as fast as possible. The target disappeared five seconds after the task started. The participants then return their hands and robots to the starting area.

Each trial consisted of eight tasks, i.e., 4 hand signs × 2 reaching positions. The order of the hand signs and positions was randomized. At the end of the eighth task, the participants were asked to take off the HMD with a text instruction.

We used a questionnaire to assess participants’ sense of body ownership and agency. The questions related to the sense of body ownership and agency in the virtual embodiment questionnaire [55] were modified and used. The questionnaire consisted of eight items in two subsets of questions for the sense of body ownership and agency, as shown in Table 2. Each response was scored on a seven-point Likert scale (1 = strongly disagree; 7 = strongly agree). The scores for the sense of body ownership and agency were calculated by taking the average of the corresponding four questions as suggested by Roth et al. [55]. In addition, task load was measured using the NASA TLX questionnaire [14]. The pairwise comparisons of the factors were performed only after the first trial.

Practice trials were conducted before the experiment to reduce learning effects. Alcohol disinfection was performed on the experimental apparatus and the hands of the experimenter and participant. Participants took a seat and were briefed on the experiment, procedures, data handling, risks, and rights and were instructed to sign a consent form if they agreed. The participants put on the HMD, and if they had no problems with the fit, they did two practice trials (one with the 30 mm, sparse, and bone-dynamic settings and another with the 30 mm, dense, and bone-dynamic). After the practice trials, the participants reviewed the questionnaires, and if they did not understand the question, an explanation was provided by the experimenter.

After the practice trials, the main experiment started. Participants wore the HMD while making the four hand signs in  Figure 5 a random order, making each hand sign twice, once toward the left side of the table and once toward the right side of the table. Then, they removed the HMD and answered the questionnaires on an iPad. The participants repeated the task of making the hand signs and questionnaire responses a total of 18 times. To control for learning effects, the order of the experimental conditions (robot size, density, and subgoal position generation and assignment algorithm) was randomized. To control the interference effect of arm fatigue, the participants were asked to ensure that they were not fatigued before each task. A five-minute break was provided after the ninth task. After the 18th questionnaire response, the participants filled out a demographic questionnaire, and a semi-structured interview was conducted for approximately five to ten minutes. The entire experiment lasted approximately 90 min.

The body ownership score, agency score, and task load index were calculated for three factors: robot size, density, and subgoal formation generation and assignment algorithm. The results are shown in Figure 7, Figure 8, and Figure 9. As these were nonparametric data, we performed an Aligned Rank Transform (ART) [75] followed by a three-factor two-way repeated-measure ANOVA with Holm correction for each subscale (i.e., body ownership, agency, and task load) to investigate the main effects and interactions.

In a VR psychophysical experiment, we studied how swarm robot factors like size, density, and algorithm impact embodiment. We focused on the sense of body ownership, sense of agency, and task load in the experimental results.

We developed custom-made swarm robots to conduct an embodiment experiment in the real world. The robot design was inspired by Zooids  [34], an open-source swarm robot platform, but the hardware and software were newly designed.

A total of 10 participants (4 males and 6 females; 28.89 ± 13.83 (SD) years old) participated in the experiment. Participants were recruited through a social media post. All the participants were unaware of the purpose of the experiment, had normal or corrected vision, and were right-handed. The participants signed a consent form regarding the experiment and were compensated with approximately $16 on Amazon gift cards. The ethics review board approved this study.

Figure 12 displays the experimental setup, which is similar to the VR experiment. In this study, a hand tracker (Leap Motion Controller 2⁸ from Ultraleap) was used to track hands instead of Meta Quest 2 to create the system without an HMD. The hand tracker is located under the table, as shown in the Figure 12 (bottom).

This experiment was similar to the VR experiment. However, the robot size was fixed at 30 mm, followed by a 3 × 3 factorial design. The independent variables examined were density (sparse, medium, and dense) and subgoal position generation and assignment algorithm (bone-static, bone-dynamic, and silhouette-dynamic). All variables were within the subject.

The task was similar to that in the VR experiment, which involved reaching a target with a specified hand shape. One difference was that in the VR experiment, the targets were positioned in the right and left fronts, but in the real-world experiment, the target position was limited to the front, and accordingly, the number of tasks per trial was halved to four (i.e., four hand signs). This is because the swarm robots’ battery capacity would not hold a charge until the end of the experiment, with eight tasks per trial.

In addition, instead of the targets automatically (dis)appearing on the desk in the VR, the experimenter manually placed and removed the targets printed on paper on the desk. To avoid the potential slipping of the robots on the target paper, the task was changed from moving the swarm robots to fit in the target to moving the swarm robots to a specified position in front of the target and making a specific hand shape.

The same subjective measurements as those used in the VR experiment (i.e., the modified embodiment questionnaire and NASA TLX) were used to evaluate the sense of body ownership, sense of agency, and cognitive load.

This procedure is similar to that used in the VR experiments. Practice trials were conducted before the experiment to reduce the learning effects. After signing the consent form, participants started the practice trials (one under sparse and bone-dynamic and another under dense and bone-dynamic).

During the main experiment, the participants wore headphones with white noise and put their hands under the table to control the swarm robots. The participants were instructed to move the swarm robots representing their right hand under the table to the starting area at the beginning of each task. When all the robots returned to the starting area, the experimenter specified a hand sign by posting handshapes on paper on the desk. The participants were instructed to move their swarm robot hand forward with a specified hand sign. The hand sign sheet was removed after five seconds, and the participants moved their hands back to the starting area. Participants repeated the reaching task and completed the questionnaire nine times.

To control the interference effect of arm fatigue, the participants were asked to ensure that they were not fatigued prior to each task. A five-minute break was provided after the fifth task. After the ninth questionnaire response, the participants answered a demographic questionnaire, and a semi-structured interview was conducted for approximately five to ten minutes. The entire experiment took approximately one hour.

The body ownership score, agency score, and task load index were calculated for each combination of robot density and subgoal position generation and assignment algorithm. The hand tracker did not work properly and frequently lost track during the experiment of one of the participants, owing to the reflection of its own infrared light at the bottom of the table. As this might have strongly affected their data, we excluded them from the results and analysis. The results are shown in Figure 13 and Figure 14. Similar to the analysis of the VR experiment results, we performed an ART procedure followed by a three-factor, two-way repeated-measures ANOVA with holm correction for the body ownership score, agency score, and task load index.

To investigate the influence of swarm robot-unique factors on embodiment in real-world settings, we conducted a psychophysical experiment with the real swarm robots we developed, analyzing three aspects: sense of body ownership, sense of agency, and cognitive load. The results were also compared with those of the VR embodiment experiment in section 4, discussing the unique characteristics of the real-world system’s effect on swarm robot embodiment.

Swarm Body can split from a single swarm into multiple swarms, each representing different body parts, and then merge back into a single swarm. These splits and mergers allow the user to adjust the number of independent swarms and the number of robots constituting each swarm as shown in Figure 15. This enables the user to seamlessly switch between one- and two-handed telepresence in a single interface. For example, when organizing a desktop remotely, the user can employ both hands to efficiently collect objects and then marge the robots to the dominant hand for precise organization. The user can duplicate one hand into two, enabling the performance of two similar tasks simultaneously through parallel embodiment as in [68]. Additionally, unlike existing embodied robots that require one system for each user, Swarm Body supports multiple users manipulating the robots through a single interface. For example, while one remote user engages in an activity such as rolling a ball with a local person, another remote user can allocate half of the robots to form a new hand and participate.

The malleability of Swarm Body enables the transformation of the swarm into various body parts of different sizes and unconventional forms (e.g., a small hand, elongated fingers, and a tentacle-like limb) (Figure 16). This transformation provides enhanced interaction freedom while preserving embodiment features, such as intuitive swarm manipulation. For example, one can experience the affordance of objects from a child’s perspective by interacting with them using Swarm Body, which simulates a smaller hand, as in [47]. Swarm Body can extend its fingers or transform them into tentacle shapes to reach and grasp objects at a distance or in narrow gaps. Similar to how pixels on a screen represent a range of embodied avatars, Swarm Body physically embodies diverse avatars through its ability to transform.

Swarm Body adapts to its environment by avoiding obstacles, resizing, or transforming itself as needed. As shown in Figure 17, our control method introduced in section 3 enables the swarm robots to not only mimic the body movements but also avoid obstacles. In a tabletop environment with obstacles, the user can interact with the environment without the need to intentionally avoid the obstacles. For example, when picking up a pen on the other side of a mug, the user can reach for it as if the mug did not exist. We believe that Swarm Body could develop into a system that seamlessly adapts to various settings, easing interactions even in cluttered spaces giving it the potential to eliminate physical barriers in interactions, enabling smoother engagements than what our own bodies can typically achieve.

Swarm Body allows the user to exert horizontal forces on remote objects or individuals in an embodied manner. This further expands the design space of physical telepresence with vertical actuation previously explored by Leithinger et al. [37]. Its embodiment aspect can also introduce intuitive and affective interactions, taking advantage of swarm characteristics in swarm user interface (SUI). Swarm Body achieves haptic communication by touching people Figure 18. Specifically, they can naturally get a person’s attention or express emotions to an intimate partner with various forces and touch. For example, by gently tapping a person’s arm engrossed in desk work, the robot can capture the person’s attention and initiate communication. Although previous studies have explored haptic feedback using swarm robots [26], our system enables haptic feedback with the embodiment of the user. Thus, Swarm Body has the potential to haptically mediate the emotions and intentions of the user to the notified person.

Another interaction modality of Swarm Body is vision. When embodied as hands, swarm robots can communicate through gestures with a physical presence in remote environments, as illustrated in Figure 19. A remote individual can utilize physical gestures during their online presentations to boost engagement. In a remote collaboration scenario, a remote user can point to specific objects or convey simple reactions. For example, a remote craft instructor can point to the tools the students need to use at each step, direct their hand movements, and send a physical thumbs-up reaction upon task completion.

Our investigation focused on a subset of the factor levels that should be examined to understand the embodiment of swarm robots on the hand. Therefore, further studies should explore other possible factor levels. For example, it is possible to investigate the embodiment characteristics of robots smaller than 20 mm in size or under even denser conditions than the dense condition in the current work. Studies in VR on these conditions may show how much embodiment is possible in theory. The obtained level of embodiment for this theoretical condition will be a baseline when evaluating real-world systems.

We did not examine some design parameters and complex conditions to make the experiments feasible. Future research should explore various parameters including the robot’s latency, speed, acceleration, color, and shape. Additionally, dynamic changes of the parameters seem to be promising approaches. For example, applying the bone-based algorithm to the fingertips and a silhouette-based algorithm to the other parts of the hand could integrate the advantages highlighted in our discussion. Also, robots in a swarm can have different sizes, shapes, densities, and functions and change them dynamically as Li et al. demonstrated [39]. Therefore, further investigation on the embodiment characteristics and applications of such swarm robots and algorithms is expected.

While we revealed the embodiment characteristics of swarm robots for the operator, we did not investigate whether an external observer could recognized the robots as someone’s body parts. In our preliminary testing, an observer was able to differentiate hand signs, although they were aware that the robots were representing a hand in advance. Since visual feedback is the only information source for an observer, a silhouette-based subgoal generation algorithm, which reduced the embodiment level for the operator in our study, might enhance their recognition of the robots as a hand. Further study on how an external observer recognizes the swarm robots is expected.

Swarm robots moving in 3D space, such as swarm drones, should be explored. Although some of our methods and findings will be applicable to them, swarm robots moving in a 3D space have unique design parameters (e.g., subgoal positions on the skin surface only vs. those on the entire volume, including the interior of the hand). This exploration also provides insight into how dimensional matching of the user’s hand movements to the robot’s subgoal formation affects a level of embodiment. If 3D formations give a higher level of embodiment, then restricting the user’s hand movements to the 2D plane (i.e., avoiding supinations and pronations) may also improve the level of embodiment in our 2D system. In addition, their 3D formation and movements will open up a further interaction design space. For example, embodied 3D swarm robots could pick up objects or enter space that is not accessible for wheeled swarm robots. Note that the projection-based robot localization method used in our study could be adapted to estimate the position and orientation of such swarm robots in 3D space [17]. These can be determined by solving the Perspective-n-Point problem, which involves considering the sensors mounted on the robots as points.

The lack of investigation on body parts besides hands also limits this study. We focused on hands as they are the most commonly used body parts for interaction with the environment. Although we revealed the embodiment characteristics of swarm robots for the hand, it remains unclear if the same tendencies apply to other body parts. We believe that our findings in the hand can be used to verify future studies on other body parts. Moreover, to expand the design space of Swarm Body, future research focusing on the effects of embodiment in body parts that differ from the user’s actual body size and shape could be highly beneficial. This would open up new capabilities and applications for Swarm Body, allowing the user to manipulate objects at micro or macro scales with larger or smaller hands or using different shapes such as tentacle-like limbs.

Although we have presented several application scenarios, their effectiveness has yet to be evaluated. In future work, we plan to showcase our applications through interactive demonstrations and videos, thereby collecting direct feedback from participants. We also expect that this feedback will allow us to discover new application possibilities of Swarm Body beyond our initial scope.

Lastly, collisions and misalignments between our robots might have affected the participants’ evaluation of embodiment. During the embodiment experiments in both VR and the real world, the experimenter sometimes observed collisions and misalignments of the robots. These undesirable behaviors potentially come from conditions, such as a large robot size or a high density, that make it difficult for swarm robots to avoid collisions while following a hand. As mentioned in section 5.8.2, our control program shuts down the robot in the case of excessive torque to protect the motors. This might affect the participants’ sense of embodiment, leading to a different result from the VR study, even though the experimenter turned them on immediately. In other words, the participants’ preference toward the denser swarm robots might be the characteristics of our real-world implementation not our approach. However, our real-world experiment and the interview responses provided some consistent and generalizable insights, such as a higher level embodiment coming from fingertip representations and a lower cognitive load in sparse conditions. To address the collision issues, a specialized swarm control algorithm tailored for embodied behavior could potentially reduce collisions and enhance the embodied experience. Further psychophysical embodiment experiments with a control algorithm with fewer collisions will help us understand whether the differences in our VR and real-world study results are due to the approach or our implementation. It will also deepen our understanding on the embodiment characteristics we observed in both VR and real-world settings (i.e., a higher level of embodiment coming from fingertip representations and a lower cognitive load in sparse conditions).

The robot receives the subgoal (target) position from the host computer via RF communication and moves to follow it. It is desirable for the robot to follow a smooth path to the subgoal position. In controlling wheeled robots, using a Bézier, spline, or cross-oid curve as the path is common. This curve passes through the current and subgoal positions to avoid sudden changes in angular velocity. However, due to the limited computational resources of the robot’s microcontroller, it has been difficult to implement control that sequentially calculates and follows these curves. On the other hand, since the robot can get its absolute position information with the projection-based method, and the subgoal position is updated by the host computer every 100 ms, the distance between the current position and the subgoal position is considered to be close, and we thought that sudden changes in angular velocity would be rare even on a path connecting these two points by a straight line. Thus, we implemented a control model that follows the straight path.

The concept of the control model is shown in Figure 20. Let P₀(x₀, y₀) be the robot’s initial position at time t = 0 given the subgoal (target) position and P_G(x_G, y_G) the subgoal position. Next, define \(\overrightarrow{P_0 P_G}\) as the vector connecting points P₀ and P_G, with L (L = L₀ at t = 0) as the distance between them, and θ (θ = θ₀ at t = 0) as the angle between the robot’s direction of motion and the vector \(\overrightarrow{P_0 P_G}\). (Counterclockwise is defined as positive.) In this case, the following law controls the velocities V_l and V_r of the robot’s left and right motors.where \(K_L, K_\theta, K_{\dot{\theta }}\) are constants that represent the control gain. Based on the above control law, \(\overrightarrow{P_0 P_G}\), the distance L_n, and the angle θ_n are updated from the robot’s current position P_n(x_n, y_n) to the subgoal position, and V_l and V_r are calculated and output. However, if this control law is followed, the point P_G is theoretically unreachable and will never fully converge, since L = 0 and the velocity will converge to 0 as the robot approaches P_G. Therefore, the robot is judged to have converged when it is within a certain distance σ from P_G, and the robot stops at that point. In addition, the minimum velocities V_{l, min} and V_{r, min} are set for V_l and V_r, respectively. The maximum speed of motors V_max is determined by the maximum speed value of the slower of the two motors. This is based on the actual measured speeds of the left and right motors as an upper limit. This can be written in the formula as follows:where V_{l, max} and V_{r, max} are the maximum speed value of left and right motors.

If a new subgoal position P_G is given before convergence, P_G is updated, and the subgoal following continues. Here, the dimension (unit) of the calculated V_l and V_r is [mm/s], but the dimension of the PWM duty of the voltage value, which is a control value that can be input to the motor, is [rpm]. Therefore, the calculated V_l and V_r are converted to the PWM duty value by the following formula:where Duty_l and Duty_r are the PWM values of left and right motors, and f(x) and g(x) are functions corresponding to the PWM value calculated by the robot’s calibration mode. We confirmed that this control rule can be used to control the robot to successfully reach the goal position even with a microcontroller with limited computational resources.