Mobile robot interception using human navigational principles: Comparison of active versus passive tracking algorithms

Transcription

1 Auton Robot (2006) 21:43 54 DOI /s Mobile robot interception using human navigational principles: Comparison of active versus passive tracking algorithms Thomas G. Sugar Michael K. McBeath Anthony Suluh Keshav Mundhra Received: 12 October 2004 / Revised: 16 January 2006 / Accepted: 10 February 2006 / Published online: 22 June 2006 C Science + Business Media, LLC 2006 Abstract We examined human navigational principles for intercepting a projected object and tested their application in the design of navigational algorithms for mobile robots. These perceptual principles utilize a viewer-based geometry that allows the robot to approach the target without need of time-consuming calculations to determine the world coordinates of either itself or the target. Human research supports the use of an Optical Acceleration Cancellation (OAC) strategy to achieve interception. Here, the fielder selects a running path that nulls out the acceleration of the retinal image of an approaching ball, and maintains an image that rises at a constant rate throughout the task. We compare two robotic control algorithms for implementing the OAC strategy in cases in which the target remains in the sagittal plane headed directly toward the robot (which only moves forward or backward). In the passive algorithm, the robot keeps the orientation of the camera constant, and the image of the ball rises at a constant rate. In the active algorithm, the robot maintains a camera fixation that is centered on the image of the ball and keeps the tangent of the camera angle rising at a constant rate. Performance was superior with the active algorithm in both computer simulations and trials with actual mobile robots. The performance advantage is principally due to the higher gain and effectively wider viewing angle when the camera remains centered on the ball image. The findings confirm the viability and robustness of human perceptual principles in the design of mobile robot algorithms for tasks like interception. Keywords Mobile robot navigation. Visual servoing. Perceptual principles T. G. Sugar ( ) M. K. McBeath A. Suluh K. Mundhra Arizona State University, Tempe, AZ Introduction Perceptual models that specify how to navigate to catch fly balls are formulated around coordinate systems that are either viewer-based or world-based (McLeod and Dienes, 1993; McBeath et al., 1995; Aboufadel, 1996). Models with viewer-based coordinate systems utilize a geometry that is relative to the vantage of the moving viewer, and only allow information that is perceptually available from that perspective. Such models can use simple control algorithms that are consistent with the behavior of humans and other locomotive animals. In contrast, models with world-based coordinate systems utilize a geometry that is independent of the vantage of the viewer, and typically must rely upon knowledge of physics. Such models either predict the target trajectory from well-defined initial conditions and known environmental factors, or triangulate the ongoing world coordinates of the target and pursuer using multiple external cameras. This approach to modeling also typically assumes knowledge of maps and landmarks of the world, information that often is not perceptually available to a viewer, and may not be reliable or possible to gather. Moreover, studies have shown that human fielders are relatively unaware of the destination of the ball during interception tasks (McBeath et al., 1995; Oudejans et al., 1996; Shaffer and McBeath, 2002). The viewer-based models appear to be more appropriate for describing action tasks in a limited information context (Milner and Goodale, 1995). Over the last decade, perceptual psychologists have proposed several viewer-based catching algorithms. One strategy called the Optical Acceleration Cancellation (OAC) model proscribes how a fielder can intercept a fly ball by selecting a running path that maintains optical acceleration cancellation of the image of the approaching ball, which results in a constant optical rate in a vertical image plane. In this

2 44 Auton Robot (2006) 21:43 54 Fig. 1 The Optical Acceleration Cancellation (OAC) strategy. A running fielder (headed from the right) approaches a projected ball (headed from the left) in equal temporal units (t 1 t 7 ). The OAC strategy directs the fielder to select a running velocity that cancels the acceleration of the image of the ball in a projection plane that moves with the fielder. In the diagram this results in a projection point moving upward along the tilted dotted line shown. This happens because, as the fielder moves forward, the distance between the fielder and the projection plane remains fixed. The fielder approaches the ball along a path such that the height of the ball in the projection plane increases by a constant amount in each fixed interval of time (d(tan(α))/dt = constant) strategy, fielders run along a path that remains aligned with the ball and maintains a constant change in the tangent of the vertical optical angle, α (see Fig. 1). This model focuses on objects hit directly at the fielder. Thus, in order to catch the ball, a fielder simply needs to select a running path that maintains the change in tanα equal to a constant, (d/dt(tan α) = c) (Chapman, 1968; McLeod and Dienes, 1993; 1996; Oudejans et al., 1996). When a fielder uses the OAC strategy, if the rate of change of tan(α) increases, it means that he is running forward too fast (or not fast enough backwards) and the ball will fall behind him. Similarly, if the rate of change of tan(α) decreases, it means he is running forward too slowly (or too quickly backwards) and the ball will fall in front of him. The rule to catch the ball is simple. The fielder can guarantee interception by selecting a running speed that maintains a constant d(tan α)/dt, or an optical acceleration cancellation (OAC) of the image of the ball. If the d(tan α)/dt equals a constant and the projection plane remains a fixed distance from the fielder, then by definition the image of the ball will rise at a constant rate. Therefore, the OAC control algorithm is very simple. If the velocity of the ball in the image plane starts to change, the fielder acts accordingly by moving forward or backward to maintain the image of the ball moving upward at a constant rate (McBeath et al., 2001; Sugar and McBeath, 2001; Suluh et al., 2001; Mori and Miyazaki, 2002; Mundhra et al., 2002; Suluh et al., 2002; Mori and Miyazaki, 2003; Rozendaal and van Soest, 2003; Miyazaki and Mori, 2004). In 1968 Chapman demonstrated that if a fielder uses OAC to approach a ball that travels through a perfect parabola, then there is an ideal solution with a straight-line constant velocity running path (Chapman, 1968). In 1985, Brancazio questioned the utility of Chapman s proposed OAC algorithm because it ignores air resistance, which substantially alters the trajectory of most projectiles (Brancazio, 1985). Thus acceleration of tan(α) of a ball affected by drag is not precisely zero when the ball is on collision course with a stationary observer. Brancazio suggested that when humans intercept a ball, they maintain a constant increment of the optical angle, α (Brancazio, 1985). But psychologists like (McLeod and Dienes, 1993, 1996), confirmed that maintaining OAC will lead to interception even when the ball trajectory is airresistance shortened, this just leads to a non-constant running speed. Moreover, they also confirmed that human fielder behavior is consistent with utilization of the OAC strategy. In 1995, McBeath et al. confirmed that fielders maintain OAC even when balls are headed off to the side (McBeath et al., 1995). In 1999, Oudejans et al. 1999) tested if the control mechanism for canceling acceleration of the tangent of the optical angle α is achieved by maintaining constant speed of the

3 Auton Robot (2006) 21: ball image along the retina, or by maintaining fixation on the ball and a constant rate of change in the tangent of the fixation angle. The former, maintaining constant speed of the ball image along the retina, requires that the perceptual system null out any changes in retinal angle and keep track of ball position relative to the background scenery. In contrast, maintaining fixation on the ball and a constant rate of change requires control and monitoring of extra-retinal mechanisms, such as the vestibular and proprioceptive systems (eye and head movements). Their findings supported the latter model, that fielders move their head and eyes and maintain fixation on the ball with constantly changing eye angle. They also challenged the need for background scenery by confirming that fielders reliably intercept luminous fly balls in the dark in the absence of visual background, under both binocular and monocular viewing conditions. They concluded that eye and head movements are used to determine the optical angle of the falling object. In the robotics literature, Borgstadt and Ferrier (2000) tested several different algorithms to intercept a flyball. In the first algorithm, the robot maintains its velocity proportional to the magnitude of d 2 (tan α)/dt 2, where α is the optical gaze angle. This strategy has difficulties when the ball is too close to the fielder and when the gaze angle approaches its maximum value resulting in unstable behavior for the robot. The second algorithm sets the fielder s acceleration equal to a constant value. In this strategy, the robot either increases or decreases its velocity depending on the sign of d 2 (tan α)/dt 2. They found that this bang-bang strategy works better. However, the implementation of this strategy by collecting acceleration data from the image is very noisy, and this bang-bang approach results in non-smooth robotic trajectories. Rozendaal and van Soest analyzed the OAC algorithm to determine when it is valid and argue that the current model will always generate a path to intercept the object for balls starting at the horizon and hit towards the fielder (Rozendaal and van Soest, 2003). They argue that other degenerate cases such as balls falling down or balls starting below the horizon cause problems for the current OAC model. In our research, it has been found that humans do have difficulty intercepting falling objects and balls launched away from them, consistent with the predictions of Rozendaal and van Soest (McBeath et al., 2002). Our work has been shown that different OAC models are used for falling balls and for intercepting balls rolling on the ground (Mundhra et al., 2002; Mundhra et al., 2003). Mori and Miyazaki have developed a robot that can intercept objects using a modified model, Gaining Angle of Gaze (Mori and Miyazaki, 2002; 2003; Miyazaki and Mori, 2004). This new research focuses on the human perceptual information, and defines viewer-based control variables that can be used in visual servoing tasks. Corrective action is taken based on simple geometrical principles. These models are interesting because the control action is taken in the visual plane and the controller is inherently different because the velocity and acceleration of the pursuer need not and does not approach zero as the destination is reached. In the next section, we define a passive and an active algorithm for implementing the OAC strategy. We then simulate and experimentally demonstrate smooth robotic controllers based on this human perceptual strategy. Modeling and simulation Two algorithms for implementing the OAC strategy are developed and compared in our research, a passive one and an active one. In the passive case, the camera on the robot is kept at a stationary angle relative to the world, whereas in the active case, the camera tilts to maintain a constant rate of the tangent of the gaze angle. The passive and the active OAC algorithms are first compared using computer simulation of robot behavior for different ball trajectories, both with and without drag. Passive OAC model In the passive model, the image of the ball on the camera must rise according to the following perceptual equation, d dt (tan α) = d ( ) yb = C (1) dt x f x b where α is the vertical optical angle of the ball from the perspective of the fielder, (x f, 0) and (x b, y b ) are the world coordinates of the fielder and ball, and C is a constant to be determined at the beginning of the task and remains fixed during the entire task. From the OAC perceptual model, it can be shown that: tan α = Ct (2) x f = x b + y b Ct where, C = ẏb(0) D, D = x f (0) x b (0) (4) D is the initial distance between the fielder and the ball and remains fixed during the task. Equation (4) is the same as Eq. (5) in Rozendaal and van Soest s research (Rozendaal and van Soest, 2003) if it is assumed that the initial height of the ball is zero. From Eq. (3), whenever the height of the (3)

4 46 Auton Robot (2006) 21:43 54 ball is zero during the task, the fielder s position is forced to converge to the ball s position. For this algorithm, the fielder or robot moves so that the image of the ball rises constantly in stationary image plane in order to achieve interception (see Figs. 2 and 3).The camera or head is fixed and does not tilt, and the image of the ball rises upward at a constant rate. The passive algorithm is implemented by using a stationary camera, and adjusting fielder position so that image of the ball rises according to the perceptual control model d/dt tan (α) = C. Because the focal distance of the camera is fixed, this constrains the distance to the projection plane, denoted by D, to be fixed so the vertical image plane effectively moves with the fielder. In the pinhole camera instantiation of the passive OAC algorithm, the height of the ball on the image plane is given by Y image. Since the focal distance is fixed, the image plane can be modeled to move forward or backward as the fielder moves forward or backward. Based on similar triangles with a fixed focal distance, d, the ball image will rise at a constant rate on the CCD image plane throughout the entire interception task (see Fig. 3). In the projection plane, the ball height equals tan(α)d, which simplifies to CDt, because tan(α) equals a constant times time. The rate of change in the height of the ball in the projection plane will equal CD = ẏ b (0). The robot estimates the initial optical velocity after sampling for 4 frames or 0.27 s. The variable, v Projection, is the robot s estimate of CD, the initial upward velocity of the projected image of the ball. By modeling the problem into a viewer-based robotics perspective, the perceptually available information in real world coordinates is converted to viewer-based image coordinates. Figure 4 (like Fig. 1) graphically illustrates Eq. (1) and how the projected image of the ball keeps rising at a constant rate given by v Projection, even while the ball physically descends near the end. Passive robot controller Our passive control law for the robot contains two terms. The firstterm,b, is the feed-forward velocity and the second term is the error between the actual height of the ball and the desired height of the ball in the projection plane, with a gain factor, K p. The desired height of the ball in the projection plane should increase at a constant rate equal to v Projection.The feed-forward term is determined by taking the derivative of Eq (1). Because B cannot be determined experimentally, it is set to zero in the simulations, but was included to allow for future simulations that can more closely mimic biological interception behavior. ẋ f = B + K p (( yb x f x b ) ) D v Projection t (5) Fig. 2 Shown is the OAC strategy in which the image height of the ball, Y projection, varies as a function of robot and ball position. The projection height is defined by a directed line through the points (x f,0)and(x b, y b ) onto a projection plane a fixed distance, D, away B = ẏ b (t f ) v Projection t f D + ẋ b (t f ) (6) Fig. 3 Shown is an enlargement of the camera geometry in the circled portion of Fig. 2 Fig. 4 The passive OAC algorithm when intercepting a fly ball is shown at a set of time intervals. The image of the ball rises at constant velocity relative to the background scenery

5 Auton Robot (2006) 21: In simulations, the robot s position will converge to the destination of the ball, x b (t f ), where t f is the landing time. Simulations were made for ball trajectories either with or without drag, and landing locations were varied to be either in front or in back of the robot s initial position. Figures 5 and 6 show the ball trajectory both without, and with drag, respectively. In these simulations, we used a controller gain of K p = 0.25, t = 1/15 (frame rate in seconds), x b = 0.5 t (m), v Projection = 2 m/s, and feedforward velocity of B = 0. In the simulations without drag, y b = 0.5 t t (m). With the perfect knowledge of v Projection and B, the robot s velocity is constant; otherwise, the robot must more quickly chase the ball toward the end of the ball trajectory. This control model allows a robot to intercept the target object on the run, similar to documented human behavior. It is different from conventional controllers where acceleration and velocity typically are designed to decrease to zero as the destination is reached. In experiments, the robot will estimate v Projection and keep it fixed during the entire task. The effects of the estimation of v Projection are shown in the next three simulations to show the robustness of the controller. For these simulations, K p = 0.25 (controller gain), t = 1/15 (frame rate in seconds), D = 3 (initial distance between the robot and the ball in meters), y b = 0.5t 2 + 2t (m), x b = 0.5t (m), and B = 0. In the simulations, three different estimates for v Projection (1,2,and3m/s) are used for the same ball trajectory and starting robot position. In one simulation, if v Projection is estimated too low, v image = 1 m/s, then the robot does not catch the ball and the robot moves backward and forward in a poor choice of running velocities. In the second two cases when v Projection = 2 and 3 m/s, the robot catches the ball, but running velocity still varies more than is desirable, in contrast to an optimal constant speed (see Fig. 7). Fig. 5 (a) Shown are robot movement trajectories for intercepting a ball without drag using the passive OAC algorithm at two initial distances (x f (0) = 1 and 3 m). Robot and ball movements are only in the x direction (plotted vertically) and shown as a function of time. The ball trajectory is indicated by a solid straight line, and the robot trajectories, starting at 1 and 3, are indicated by dotted lines. (b) Shown are the heights of the projection of the ball as a function of time using the passive OAC algorithm for the two trajectories in (a). The desired projection height of the ball is indicated by a solid, straight line. The simulated ball projections on the projection plane for initial robot positions at 1 and 3, respectively, are indicated by dotted lines that curve somewhat initially, but remain close to the linear ideal (c). Shown is the velocity of the robot starting at two initial distances (x f (0) = 1and 3 m). The ball is headed to land behind the robot starting at an initial distance of 1 m, which has a negative (backward) velocity that stabilizes after about 1.5 s. The ball is heading in front of the robot starting at an initial distance of 3 m, which has a positive velocity that continues to increase until near interception. A more optimal solution would necessitate earlier acceleration to achieve a more constant velocity Active robot controller In the active OAC algorithm, the tilt of the camera on the robot is constantly adjusted using the formula: α des = arctan (Ct) (7) The robot estimates C by setting it equal to the initial observed v Projection /D and tilts the camera based on the arctan function. Here, the desired projection of the ball is always at the center of the projection image. When the ball is not at the center of the projection image, the robot corrects the error by moving forward or backward. It is assumed that the pinhole is fixed and the image rotates about the pinhole. In this case, a similar formula is derived. The function in the

6 48 Auton Robot (2006) 21:43 54 denominator occurs because the calculation is performed in a rotating image plane that is perpendicular to the gaze direction. (( ) )/ yb ẋ f = K p D v Projection t ( 1 + x f x b )( )) vprojection t ( yb x f x b D (8) Fig. 6 (a) Shown are the robot movement trajectories for intercepting a ball with drag using the passive OAC algorithm at two initial distances (x f (0) = 1 and 3 m). The drag is given by 0.05 v 2. The ball trajectory is indicated by a solid line, and the robot trajectories starting at 1 and 3 m are indicated by dotted lines. (b) Shown are the heights of the projection of the ball with drag as a function of time using the passive OAC algorithm for the two trajectories in Fig. 6.(a). The desired projection height of the ball is indicated by a solid, straight line. The simulated ball projections on the projection plane for initial robot positions at 1 and 3 m are indicated by dotted curved paths. (c) Shown is the velocity of the robot starting at two initial distances (x f (0) = 1and3m).The ball is headed to land behind the robot starting at an initial distance of 1 m, which has a negative (backward) velocity that stabilizes after about 1.5 s. The ball is heading in front of the robot starting at an initial distance of 3 m, which has a positive velocity that continues to increase until interception. The terminal velocity at 3 m exceeds that without drag shown in Fig. 5.(c). as the robot moved further forward over a shorter period of time Matching the passive algorithm tests, two sets of simulations, both without and with drag, were performed. The active model simulations used values of v Projection = 2m/s,K p = 5, t = 1/15 (s), and x b = 0.5 t (m). In the simulations without drag, y b = 0.5 t t (m). As before, the robot starts from two different initial positions, at x = 1 m and x = 3m, and the ball travels with a perfect parabolic trajectory (see Fig. 8). In the simulations with drag, the robot and ball have the same initial conditions, but the modeled trajectory is shortened due to the addition of drag (see Fig. 9). With an active controller, the gain can be increased and allows the robot to maintain an error that is well under an order of magnitude smaller than for the comparable passive algorithm shown in Fig. 5. With the active controller, the robot achieves very efficient, near constant-velocity behavior and is successful in catching the ball despite starting from different initial positions, and the disturbance due to drag. The trajectories are also closer to a constant ideal (i.e. straight lines) than the comparable simulation with the passive algorithm shown in Fig. 6. The effects of the estimation of v Projection with the active OAC algorithm are shown in three more simulations without drag present. In these three simulations, K p = 5, t = 1/15 (s), D = 3(m),y b = 0.5t 2 + 2t (m), and x b = 0.5t (m). The results of the simulations are shown in Fig. 10 for v Projection = 1, 2, and 3 m/s. These rates are below, at, and above the initial image velocity that would be observed from the perspective of the robot located at the given starting distance. The robot successfully intercepts the ball in all three conditions, recovering from non-optimally varying its position based on high and low values of v Projection. The success indicates more robustness than the comparable passive algorithm shown in Fig. 7. The results of our simulations support that the active OAC algorithm is superior to the passive one. In the active model, the controller gain, K p, can be increased greatly (in this case by a factor of 20), so that the robot has better response characteristics. This leads to the error between the desired and the actual projection of the ball being markedly smaller in the active model. Thus, the velocity of the robot remains lower in the active model. As our graphs consistently show, a robot achieves more efficient solutions using the active

7 Auton Robot (2006) 21: Fig. 7 (a) Shown are three robot movement trajectories for interception using the passive OAC algorithm (v Projection = 1, 2, and 3 m/s). The trajectory of the ball in the x direction is plotted as a solid straight line, and the robot trajectories for v Projection are plotted as dotted curved paths. For v Projection = 1 m/s, the robot initially heads away in the wrong direction and in the end fails to intercept the ball. (b) Shown is the height of the projection of the ball as a function of time using the passive OAC algorithm for the three v Projection conditions shown in Fig. 7.(a). The desired projection heights are plotted in solid lines, while the actual simulated ball projections are plotted as dotted lines. For v Projection = 1 m/s, the projection height of the ball drops to zero (ground level) at the end; hence, the robot does not catch the ball. (c) Shown is the velocity of the robot starting at an initial distance of 3 m with the different conditions of v Projection. The ball is heading in front of the robot, and the robot generally has a positive velocity to move forward and quickly intercept it. In the case of v Projection = 1m/s,the initial robot velocity is negative, and then it increases greatly near the end of the simulation, trying to recover Fig. 8 (a) Shown are the ball and robot trajectories using the active OAC algorithm (x f (0) = 1 and 3 m). The ball trajectory in the x direction (solid line) and the robot trajectories (dotted lines) are plotted against time. The robot trajectories are closer to a constant ideal (i.e. straight lines) than the comparable simulation with the passive algorithm shown in Fig. 5. (b) Shown are the optical ball projections using the active OAC algorithm. The desired projection of the ball on the projection plane is a straight horizontal line. The dotted curves are the ball projections as viewed by the robot with initial positions at 1 and 3 m. (c) Shown are the robot velocities using the active OAC algorithm. The velocities rapidly asymptote to near ideal constant speeds that remain well under those achieved with the comparable passive algorithm shown in Fig. 5 OAC algorithm, and exhibits more robustness in being able to catch balls within a larger range of values of v Projection.In contrast, a robot using the passive OAC algorithm tends to vary more in velocity and fails to catch the ball when v Projection is extreme (e.g. v Projection = 1 m/s), but was catchable in the active case.

8 50 Auton Robot (2006) 21:43 54 Experiment and discussion of the results Experiment set up In our experiments, we use a Nomad Super Scout robot (Nomadic Technologies Inc.) with an additional pan-tilt mechanism (Directed Perception Inc.). The robot has an on board computer with a second computer for image processing that has a FireWire r interface (see Fig. 11). The commands to control the drive wheels of the robot, the pan-tilt mechanism, and the frame grabber are written in C language under the Linux operating system. A balloon with a light weight is used to create different slow motion trajectories of a ballistic object. The robot control program calculates the center of the balloon using a standard center of mass calculation at rate of 15 frames/s in the 2D camera image. In the first set of trials, the balloon travels in a near parabolic trajectory headed to land in front of the robot. In a second set of trials, the balloon travels in a near parabolic trajectory headed to land beyond the robot. In both cases, the robot adjusts its velocity to intercept the balloon. Pretest validating OAC strategy with stationary robot As a pretest, the OAC strategy is validated for three trials of a ball lofted along approximately parabolic trajectories. The robot is placed at the destination position of the ball and remains stationary during the test. The images captured by the robot confirm the prediction of the OAC model. Essentially constant increments are maintained in the vertical position of the center of the ball on the image plane between two consecutive frames (see Fig. 12). Experimental result of passive vs. active OAC algorithms In the first comparison between passive and active algorithms, the ball heads to land in front of the robot. For the passive algorithm, the camera remains stationary during the experiment, and the desired rise in optical ball velocity, v Projection, is set to maintain the velocity observed during the first four frames. v Projection is measured in pixels/s and could be converted to distance/s by multiplying by the focal length of the camera. It remains fixed during the entire task. In this trial, the robot establishes a desired rise velocity that is somewhat lower than optimal, which leads it to actually back up initially, trying to slow down the observed optical velocity. During the first three-quarters of the trial, the robot successfully keeps the centroid of the ball image moving upward at a near constant rate. During the last quarter, near the end of the trial, the difference between the desired and actual image Fig. 9 (a) Shown are ball and robot trajectories using the active OAC algorithm (x f (0) = 1 and 3 m) with drag present. The ball trajectory in the x direction (solid line) is plotted vertically against time. (b) Shown are ball projections using the active OAC algorithm with drag present. The robot, starting from two different initial positions, is able to maintain the image of the ball very near the center of view and intercept it. The desired image of the ball (solid line) is at the center of the projection plane. (c) Shown is the velocity of the robot starting at the two initial positions. The velocities rapidly asymptote to near ideal constant speeds that remain well under those achieved with the comparable passive algorithm shown in Fig. 6 plane locations of the centroid increases, and the robot is forced to rapidly adjust, accelerating forward quickly (See Fig. 13). The robot s velocity is negative at the beginning and then positive at the end indicating a less than optimal choice of v Projection (2 instead of 2.5) that leads the robot to initially move back and then have difficulty recovering at the end. Because the gain is set low, it is difficult not to have some error when the robot establishes the initial desired rise velocity, and the compounding of this error over time leads

9 Auton Robot (2006) 21: Fig. 11 Nomad Super Scout robot with a monocular camera mounted on a pan tilt platform. The secondary laptop is used for image processing Fig. 12 Shown is the ball centroid versus frame, as measured from a stationary robot located at the destination position for three lofted balls. The centroid of the ball rises at a relatively constant rate on the image plane, consistent with the prediction of the OAC strategy Fig. 10 (a) Shown are robot trajectories using the active OAC algorithm (v Projection = 1, 2, and 3), from a starting distance of three meters. The ball trajectory is indicated by a solid line, and the robot trajectories are indicated by dotted lines for the each of the three values of v Projection. (b) Shown are ball projections using the active OAC algorithm. The desired projection of the ball is a straight line. The actual ball projections are plotted as dotted lines respectively. The robot controlled with v Projection = 1 m/s is able to intercept the ball using an active algorithm, while failing to intercept the ball using a passive one shown in Fig. 7.(a). Shown are the velocities of the robot starting at an initial distance of 3 m. The ball is heading in front of the robot, and the robot has a positive velocity to move forwards and quickly intercept it. In the case of v Projection = 1 m/s, the robot moves backward and then forward to intercept the ball. Even in the cases when v Projection is artificially high or low, the robot velocity still asymptotes to a near constant speed before interception the robot to move non-optimally for much of the duration of the trial. For the active algorithm, the camera is adjusted so that it continuously changes in tilt at a constant rate proportional to the initial rate that the centroid of the ball rises in the image plane during the first four frames (see Eqs. (7) and 8). The robot moves backward or forward to maintain the actual centroid of the ball at the center of the image plane. As found in the simulations, the difference between the desired centroid and the actual centroid is very small. The robot moves with a continuous, much slower velocity and achieves better accuracy than it did with the passive algorithm (see Fig. 14). By comparison, the actual and the desired centroid of the ball in the passive algorithm trial (Fig. 13) are further apart which causes the robot to move with a higher velocity, particularly near the end. In the second experimental comparison, the ball moves along an approximately parabolic trajectory that is headed to land beyond the robot. With the passive algorithm, the image of the ball initially rises too fast, accelerating upward, and thus, the robot rapidly moves backward. Eventually, the robot overreacts and its inertia carries it too far causing the centroid of the ball in the image plane to dramatically decelerate and actually slow down (the ball in the image starts to curve downward), which in turn causes the robot to dramatically

10 52 Auton Robot (2006) 21:43 54 Fig. 13 (a) Shown are the desired and actual optical paths for the passive OAC algorithm with a ball that is headed to land in front of the robot (Case 1). The desired (dashed) and actual (solid) optical paths of the centroid of the ball are plotted against the image frames. The downward dip at the end indicates that the robot is having difficulty approaching the ball fast enough to keep it from falling in front. (b) For the passive OAC algorithm (Case 1), the actual velocity function (solid) and the desired velocity function (dashed) are shown for a ball headed to land in front of the robot. The initial negative velocity indicates that the robot is actually drifting backwards in the wrong direction. The curve up at the end shows the robot lurching forward, making up for its earlier inaccuracies decelerate (see Fig. 15). In the active algorithm, the robot is able to track the ball with much better accuracy, which results in a path to the ball with a much more constant, and hence slower velocity, consistent with the results found in the earlier simulations (see Fig. 16). The robot moves with a maximum of 0.15 m/s using the active algorithm and 0.51 m/s using the passive one, and the active robot velocity is much closer to optimal constancy, especially near the end. Fig. 14 (a) Shown are the desired and actual optical paths for the active OAC algorithm with a ball that is headed to land in front of the robot (Case 1). The desired centroid of the ball (dashed) for the active algorithm is at the center of the image. After 5 frames, the actual image of the ball drops (solid) and the robot moves forward (see Fig. 14.(b)), re-establishing the actual image close to the center. (b) For the active OAC algorithm (Case 1), the desired (dashed) and actual (solid) velocity of the robot increase after 5 frames in response to the drop in the actual image of the ball. The robot moves forward to catch the ball, and achieves an approach speed that gradually diminishes. The overall pattern of motion is smoother and more optimal than with the passive algorithm shown in Fig. 13 Conclusion We investigated and tested the human-based perceptual navigation principle known as Optical Acceleration Cancellation, or OAC. OAC guides a fielder to interception of ballistic objects by keeping the optical image of the approaching projectile constantly rising throughout the entire task. Mobile robotic control algorithms were developed for

11 Auton Robot (2006) 21: Fig. 15 (a) Shown are the desired and actual optical paths for the passive OAC algorithm with a ball that is headed to land beyond the robot (Case 2). The desired (dashed) and actual (solid) images of the center of the ball are plotted against the image frames. For most of the trajectory, the actual image of the ball rises at a higher rate than desired indicating that the ball continues to head beyond the destination of the robot. Eventually, after 30 frames, the robot has accelerated to the point of overcompensating, and the ball image rapidly descends, causing the robot to change directions and accelerate forward. (b) For the passive OAC algorithm (Case 2), the actual (solid) and desired (dashed) velocity of the robot correspond with the image plane data shown in Fig. 15.(a). The robot moves backward since the ball heads behind the robot s initial position. The robot initially moves backward and continues to accelerate backwards until 30 frames. At that point, the robot has over-committed in the backward direction, so the image of the ball drops in the image plane, and the robot reverses direction and moves forward to catch it the OAC strategy that has been proposed by perceptual psychologists, and robotic simulations and experiments were performed. In developing the control algorithms, we showed that keeping the derivative of the tangent of the optical angle equal to a constant is functionally equivalent to keeping the Fig. 16 (a) Shown are the desired and actual optical paths for the active OAC algorithm with a ball that is headed to land beyond the robot (Case 2). The actual (solid) and desired (dashed) centroid of the ball in the image plane are plotted against the image frames. After about 15 frames, the robot establishes a fairly regular error pattern that causes it to smoothly move backwards. (b). For the active OAC algorithm (Case 2), the desired (dashed) and actual (solid) velocity of the robot gradually increases backwards after about 15 frames (in response to the actual image of the ball being too high). Overall, the robot moves backward with a much slower and smoother velocity compared to the passive algorithm shown in Fig. 15 ball rising at a constant rate in a camera with a fixed focal length. We also demonstrated from the simulation and the experimental data that both the passive and the active OAC algorithms can succeed in guiding a robotic fielder to intercept ballistic objects. Simulations and actual robotic tests confirmed this with a variety of different paths, headed both in front of and beyond the fielder, and both without or with drag due to air resistance. We found that the active OAC algorithm, in which a camera is continually tilted and a robot adjusts its position to center the ball in the image, performs better than the

12 54 Auton Robot (2006) 21:43 54 passive OAC algorithm, in which the ball image constantly rises along a stationary camera image. In the active case, the error between the desired image of the ball and the actual image of the ball was consistently smaller than matched trials using the passive algorithm. Because the error was small, the robot s velocity was much more constant and considerably lower with the active algorithm. Moreover, the simulations indicated that the active OAC algorithm was more robust, working over a wider range of different values of v Projection, the estimation of the initial upward optical velocity of the target object. The comparison of passive and active control algorithms supports that the active method, demonstrated to be used by humans, is also superior for projectile interception by mobile robots. Moreover, our findings confirm that these type of control algorithms, ones that use viewer-based perceptual principles and visual-servoing of image data, provide a simple and robust method for mobile robots to intercept projectiles with varying trajectories. These algorithms are quite different than conventional controllers because camera calibration is not needed, only viewer-based information is needed, and the velocity of the robot does not converge to zero when it reaches the destination. Simple geometric principles are easily programmed in the visual plane and can guide autonomous robots to naturally navigate. References Aboufadel, E A mathematician catches a baseball. American Mathematics Monitor, 103: Borgstadt, J.A. and Ferrier, N.J Interception of a projectile using a human vision-based strategy. In IEEE Int. Conf. on Robotics and Automation. Brancazio, P.J Looking into Chapman s Homer. American Journal of Physics, 36: Chapman, S Catching a baseball. American Journal of Physics, 36:868. McBeath, M.K., Shaffer, D.M. et al How baseball outfielders determine where to run to catch fly balls. Science, 268: McBeath, M.K., Sugar, T. et al Comparison of active versus passive ball catching algorithms using robotic simulations [Abstract]. Journal of Vision, 1(3):193a. McBeath, M.K., Sugar, T.G. et al Human and robotic catching of dropped balls and balloons. Journal of Vision [Abstract], 2(7):434. McLeod, P. and Dienes, Z Running to catch the ball. Nature, 362(6415). McLeod, P. and Dienes, Z Do fielders know where to go to catch the ball or only how to get there? Journal of Experimental Psychology: Human Perception and Performance, 22(3): Milner, A. and Goodale, M The visual brain in action. Oxford Psychology Series 27, Oxford University Press. Miyazaki, F. and Mori, R Realization of ball catching task using a mobile robot. In IEEE Int. Conf. on Networking Sensing and Control. Mori, R. and Miyazaki, F Examination of human ball catching strategy through autonomous mobile robot. In IEEE Int. Conf. on Robotics and Automation. Mori, R. and Miyazaki, F GAG (Gaining Angle of Gaze) strategy for ball tracking and catching Task-implementation of GAG to a mobile robot. In 11th Int. Conf. on Advanced Robotics. Mundhra, K., Sugar, T.G. et al Perceptual navigation strategy: A unified approach to interception of ground balls and fly balls. In IEEE Conf. on Robotics and Automation. Mundhra, K., Suluh, A. et al Intercepting a falling object: Digital video robot. In the Procedings of the IEEE Int. Conf. on Robotics and Automation. Oudejans, R.R., Michaels, C.F. et al The relevance of action in perceiving affordances: Perception of catchableness of fly balls. Journal of Experimental Psychology: Human Perception and Performance, 22(4): Oudejans, R.R., Michaels, C.F. et al Shedding some light on catching in the dark: Perceptual mechanism for catching fly balls. Journal of Experimental Psychology: Human Perception and Performance, 25(2): Rozendaal, L.A. and van Soest, A.J Optical acceleration cancellation: A viable interception strategy? Biological Cybernetics, 89(6): Shaffer, D.M. and McBeath, M.K Baseball outfielders maintain a linear optical trajectory when tracking uncatchable fly balls. Journal of Experimental Psychology: Human Perception and Performance, 28(2): Sugar, T. and McBeath, M.K Spatial navigational algorithms: Application to mobile robotics. In Vision Interface Annual Conf., VI Suluh, A., Mundhra, K. et al Spatial interception for mobile robots. In IEEE Int. Conf. on Robotics and Automation. Suluh, A., Sugar, T. et al Spatial navigation principles: Applications to mobile robotics. In IEEE Int. Conf. on Robotics and Automation. Thomas Sugar works in the areas of mobile robot navigation and wearable robotics assisting gait of stroke survivors. In mobile robot navigation, he is interested in combining human perceptual principles with mobile robotics. He majored in business and mechanical engineering for his Bachelors degrees and mechanical engineering for his Doctoral degree all from the University of Pennsylvania. In industry, he worked as a project engineer for W. L. Gore and Associates. He has been a faculty member in the Department of Mechanical and Aerospace Engineering and the Department of Engineering at Arizona State University. His research is currently funded by three grants from the National Sciences Foundation and the National Institutes of Health, and focuses on perception and action, and wearable robots using tunable springs. Michael McBeath works in the area combining Psychology and Engineering. He majored in both fields for his Bachelors degree from Brown University and again for his Doctoral degree from Stanford University. Parallel to his academic career, he worked as a research scientist at NASA Ames Research Center, and at the Interval Corporation, a technology think tank funded by Microsoft co-founder, Paul Allen. He has been a faculty member in the Department of Psychology at Kent State University and at Arizona State University, where he is Program Director for the Cognition and Behavior area, and is on the Executive Committee for the interdisciplinary Arts, Media, and Engineering program. His research is currently funded by three grants from the National Sciences Foundation, and focuses on perception and action, particularly in sports. He is best known for his research on navigational strategies used by baseball players, animals, and robots.