Analytical Longitudinal and Lateral Models of a Spherical Rolling Robot

Transcription

1 Analytical Longitudinal and Lateral Models of a pherical Rolling Robot Jean-François Laplante, Patrice Masson and François Michaud Abstract Roball is a spherical robot designed to be a platform to study child-robot interaction in open settings. It has a spherical shell and uses motors attached along its rolling axis plus an actuated counterweight for steering. It is designed to be inexpensive and to generate a wide variety of movement. To evaluate its locomotion capabilities in relation to the size and weight of its components, we have derived analytical models describing its longitudinal and lateral motion. imulation results are presented using MATLAB/imulink and immechanics, and validated using a real robot. I. INTRODUCTION spherical rolling robot is a mobile platform that moves Aby using a mechanism that either change the center of gravity or generate a force to make the robot roll on its outer shell. pherical rolling robots can be categorized into six categories: A single driving wheel placed at the bottom inside a sphere, attached to axis supporting a controlling box, a spring and a balance wheel [1][2]. A spherical wheel with an arch-shaped body outside the sphere, similar to a mono-cycle robot [3]. A hollow sphere with a small car [4] resting on the bottom of the sphere. The concept is similar to having a gerbil run inside a spherical plastic ball. A sphere with two side rotors, mutually perpendicular, plus one rotor attached to the bottom of the sphere [5]. The change in angular velocities of these internal rotors turning on themselves makes the robot move. A sphere with four inertial wheels [6][7] or masses that move [8] over four axes in a tethrahedral structure. A sphere with motors attached to the side of the sphere at the position of the rolling axis. A rotational mass [9] or an actuated counterweight [1] [11] can be used to steer the robot. urveys of ball-shaped and spherical robots are presented in [2][4][12]. This work was supported in part by the Canada Research Chair (CRC), the Natural ciences and Engineering Research Council of Canada (NERC) and the Canadian Foundation for Innovation (CFI). Jean-François Laplante and Patrice Masson are with the Department of Mechanical Engineering of the Université de herbrooke, Québec CANADA J1K 2R1 ( {Jean-Francois.Laplante, Patrice.Masson}@Uherbrooke.ca). F. Michaud holds the Canada Research Chair in Mobile Robotics and Intelligent Autonomous ystems. He is with the Department of Electrical Engineering and Computer Engineering of the Université de herbrooke, Québec CANADA J1K 2R1 (phone: x 6217; fax: ; Francois.Michaud@Uherbrooke.ca). Fig. 1. First (left) and second (right) prototypes of Roball. Our robot, named Roball [1][11][13][14], fits in the last category. Roball is made of three rigid bodies: its spherical outer shell, its internal structure holding the different internal components of the robot (motors, microcontroller and circuits), and its counterweight. We built two prototypes of Roball: Roball-1 is made of a plastic sphere (bought in a pet store) attached at the middle, and Roball-2 is designed to be manufactured by thermoforming and was fabricated using Rapid Prototyping in AB (acrylonitrile-butadiene-styrene copolymers) plastic. hown in Fig. 1, both are about 6 inches in diameter. Propulsion motors are located on the side of the internal structure, perpendicular (on the horizontal plane) to the front heading of the robot and attached to the extremities of the spherical shell. They move the center of gravity of the robot forward or backward, for longitudinal motion of the robot. The speed of the motors is regulated according to longitudinal inclination of the internal structure, keeping the center of gravity of the robot close to the ground. Lateral motion is achieved using the battery as the counterweight, mounted on a servo-motor. This allows the robot to tilt on one side or the other as the shell rolls. ince cost is an issue in designing a toy robot for childrobot interaction [13], we used inexpensive sensors such as tilt sensors (first prototype using two propulsion motors resolution approximately 3 over 16 ) or custom-designed inclinometers using small pendulums, wheel encoders and photodetectors (second prototype using one propulsion motor resolution 1 over 36 ) to provide control measurement for velocity control of the robot s motors. uch increased range is required with Roball-2 because its steering mechanism allows the robot to flip over from one side to the other, over 25, as shown in Fig. 2. Note that with one of its side being flat (the other is the robot s face), Roball-2 can lie with its face facing upward, to facilitate interaction with people. Even though the six categories of spherical rolling robots have implemented prototypes, not all have detailed

2 mathematical models of the robot s motion capabilities. In relation to other types of spherical rolling robots, only [3] and [4] present models for longitudinal motion that can show similarities with Roball, but following different locomotion principles. For a spherical robot like Roball, to our knowledge, no kinematic and dynamic models of longitudinal or lateral motions have yet been developed. For our work, advanced models would not be necessary to derive a controller for the robot because of the low-precision sensors used onboard. imple models are desired to evaluate the influence of the robot s weight and diameter on its motion. the point of contact of the axis, being the length of the pendulum s shaft, g is the gravitational acceleration, and being the longitudinal acceleration of the sphere. R X + m Q = m Q cos Q 2 Q sin Q + x Q R Y g Q sin Q + 2 Q cos Q + (1) The sum of moments on the pendulum is presented by (2), with C being the torque applied at the center of the sphere and J Q the moment of inertia of the pendulum about an axis perpendicular to the X-Y plane. C m Q gsin Q = J Q Q (2) x Fig. 2. Roball-2 moving from one side to another, by moving its internal counterweight. The middle picture illustrates how Roball-2 can lie face up. In this paper, we present models that describe mathematically the robot s longitudinal and lateral motion capabilities on flat surfaces, first described using MATLAB/imulink in ection II, and then validated using immechanics and experimentally using Roball-2 as reported in ection III. II. MOTION MODEL A. Longitudinal Motion Let us start with a simplified model, only considering noslip longitudinal motion on flat surfaces. Fig. 3 illustrates the simplified model with a side view of the robot. It represents the spherical outer shell with its center of mass CM and a pendulum (composed of a massless link and a point mass at its end) with center of mass CM Q and the axis attached at the center of the sphere. Fig. 4. Forces and moments for the pendulum (left) and the sphere (right). The sum of forces and the sum of moments for the sphere are given by (3) and (4) respectively, with f representing the contact force, N the normal force with the ground, being the radius of the sphere, J the inertia of the sphere about an axis perpendicular to the X-Y plane, and the angular acceleration of the sphere. R X + m R Y g + f x N = m (3) C + f = J (4) The longitudinal acceleration of the sphere can be expressed in terms of the angular acceleration of the sphere according to (5). x = (5) Fig. 3. implified model for longitudinal motion, side view of the robot. Dynamic models can be derived for the pendulum and the sphere using Newton-Euler method, summing the forces and summing the moments. Fig. 4 illustrates the forces and moments on each of the two rigid bodies. The sum of forces on the pendulum expressed in relation to X and Y reference frame is given by (1), with R X and R Y the reacting forces at Fig. 5. Q (left) and (right) in relation to time using the non-linear longitudinal model, with m =.448 kg, =.889 m, J = 2.35x1-3 kg m 2, m Q =.5 kg, =.317 m, J Q = 5.25x1-4 kg m 2.

3 Fig. 5 illustrates Q and with respect to time with a step excitation of C =.1 Nm, using (1) to (5) is generated in a MATLAB/imulink simulation. The results indicate that the system oscillates around a mean value of Q =.7 rad, and such oscillations have an impact on the speed of the sphere. In order to simulate Roball s behavior, a model of its propulsion actuator, i.e., a DC motor, must be derived. Equations (6) and (7) present this model, with L the motor s inductance, R its resistance, k E the back electromotive force (back-emf) constant, k M the torque constant constant, Z the gear ratio, and i and e the current and applied voltage respectively. L i + Ri + k E Z( Q )= e (6) C = k M Zi (7) Considering only the forces in the X axis and using these seven equations, three non-linear equations are derived expressing the longitudinal dynamic model of the robot. Equation (8) is obtained by isolating R X in (1) and (3), isolating f using (4) replacing x by (5) and replacing C by (7). Equation (9) is derived from replacing (7) in (2). Equation (1) is derived directly from (6). m Q (r Q Q cos Q r Q 2 Q sin Q r ) + k M Zi J m r = (8) J Q Q + m Q gsin Q k M Zi = (9) L i + Ri + k E Z( Q ) e = (1) Using (8), (9) and (1), a linear model is expressed with state variables of the form x & = A x + Bu, with u equals to e, is given by (11) and (12). The model is linearized at Q = and Q =. 1 m 2 Q r 2 Q g G Q = 1 m Q g J Q i Q k E Z L k E Z L m Q k M +k M ZJ Q G k M Z J Q R L Q + e Q i 1 L (11) ( ) (12) G = J Q m Q r 2 + m r 2 + J Using the linear longitudinal model, we can demonstrate that the system is controllable by state feedback [14] in order to remove the oscillations shown in Fig. 5. However, with Roball-2, to minimize the cost, we only have onboard inclinometers installed on the robot, a small microcontroller and no velocity sensors measuring the sphere s speed of rotation. It is therefore impossible to implement a complete state-based control approach with Roball-2. The control strategy is thus to assign a constant speed setpoint for the sphere, and to use the angle Q of Roball-2 s internal structure measured by the inclinometer as the error signal. At constant velocity and without considering friction, Q. Fig. 6 illustrates the pole-zero map of the open-loop transfer function of the system, derived using the linear model given by (11). It shows an angular frequency of 3.85 rad/s and a damping of.89. For a unit step input, the gain of the steady state response would be Fig. 6. Pole-zero map (left) and unit step input (right) of the linear longitudinal model, with k E = x1-3 V/rad, k M = 2.13x1-3 N m/a. Fig. 7. Effect of K BF on the pole-zero map. In a closed-loop configuration, a proportional controller with gain K BF can stabilize Q around the desired position. Fig. 7 shows the effect of K BF on the pole-zero map. As K BF increases, the conjugated pole is moving away from the imaginary axis while the real pole is moving closer, and eventually the system becomes over-damped. K BF =.126 is the smallest gain for which the system remains over-damped and stable. This results for a system with a rise time of.62 sec, no overshoot, a natural frequency of 4.2 rad/s and a damping of 1, as shown in Fig. 8. Fig. 8. Pole-zero map (left) and unit step input (right) of the linear longitudinal model with K BF =.126.

4 B. Lateral Motion Lateral motion of Roball is achieved by changing the angle of Roball-2 s counterweight B. The first interesting aspect to model is how much the counterweight must be inclined to create lateral motion of the robot when lying on is side. Fig. 9 illustrates the position of the center of mass of the robot. To make the robot change direction, the minimal angle B of the counterweight in relation to the internal structure is given in (13). o, with l = 1.27 cm (the length of the flat surface on Roball-2 s side), B must be at least greater than 59 to make the robot change direction. A step command with B = 59 will however cause oscillations, and it would be preferable to use a ramp as a command for steering. Fig. 9. Lateral forces on the robot. B = arcsin m TOT l 2m B r B (13) Fig. 11. Lateral forces and moments on the robot. Fig. 11 shows the forces influencing the lateral motion of the robot. The sum of moments at the point of contact of the sphere with the ground is given by (16), with f C the centripetal force on the center of mass of the robot, which is also expressed by (17). f C y CM (m + m P + m B )gx CM = (16) f C = m TOT ( ) 2 r C (17) From Fig. 11, we can express B by (18), for ( /r C < 1). B = arctan r r C r C (18) Using (14), (15), (17) and (18) in (16), and knowing that m TOT = m + m P + m B = m + m Q, the radius of curvature r C in the motion of the robot is presented by (19). r C = + m ) r + m (r r )]( P B B ) 2 + m B r B g m B r B B g (19) III. IMMECHANIC AND EXPERIMENTAL REULT A. Longitudinal Motion on Flat urfaces Fig. 1. Geometry of lateral motion (back view). The other type of lateral motion that can be modeled is circular motion. Fig. 1 illustrates the centers of mass, the angles and the radius of the different rigid bodies of the robot. When B T is small, sin( B T ) B T and cos( B T ) 1. The coordinates of the center of mass of the robot, x CM and y CM, are given by (14) and (15). x CM = m B r B ( B T ) m TOT (14) y CM = (m P + m ) + m B ( r B ) m TOT (15) imulating the model described in ection II.A with MATLAB/imulink allowed to gain a better understanding of the behavior of the robot under a simple configuration with linear dynamics. However, in order to simulate the behavior of the robot under more complex configurations (inclined surfaces, with obstacles, for lateral or circular motion), we used immechanics, a imulink module that allows modeling the dynamics of multiple rigid bodies, connected together by joints and having to respect certain constraints [15]. The implementation is given in [14], considering the three rigid bodies of Roball-2. The result is similar to Fig. 8, with a rise time of 1.9 sec and with small oscillations caused by the precision of the inclinometers (set to 2.5 in the simulation). We also experimentally validated Roball-2 s behavior in such conditions, in open-loop and closed-loop longitudinal control. To do so, we installed a camera on the ceiling of the room, filmed the trajectories of the robot and used a colorsegmentation algorithm to track the center of the sphere and reconstruct these trajectories [14]. Using the positions of the

5 robot as a function of time, the velocity of the sphere can be derived. Fig. 12 shows all of the results obtained with a velocity setpoint of 4 rad/s with Roball-2 and with immechanics. Compared to the simulation, more oscillations occur with the real robot. This is caused by the fact that the resolution of the inclinometer used on the robot is 1. To avoid inadequate behavior, inclinometer data between ±1 are set to. This limitation causes oscillations but, despite that, the closed-loop control allows reducing by 6% the oscillations in the angular speed of the robot, compared to the open-loop configuration. Fig. 14. Circular trajectories made by Roball at (, B ) = (3 rad/s, 45 ). Fig. 12. Unit step input of Roball in open and closed loops with K BF =.126 using immechanics. B. Lateral Motion Fig. 13 shows results obtained using this model for different rotational velocities. The round dots represent results obtained using immechanics for (, B ) = (2 rad/s, 9 ) and for (, B ) = (4 rad/s, 24 ). MATLAB/imulink simulation results are shown for 2 rad/s, 3 rad/s and 4 rad/s. Using the ceiling camera setup, we validated some of these results using Roball-2 at =3 rad/s and B set to 15, 3 and 45. Fig. 14 illustrates one trial. It is worth noticing that the center of the circular trajectories moves as the robot turn. This is caused by low-resolution readings of the inclinometer (used in a PI controller of the angle of the counterweight), unbalanced masses in the robot and small inclination of the floor. Considering each complete circle made by the robot and averaging the radius of curvature, results in Fig. 13 shows good similarities between the mathematical model and the trajectories of the real robot. Finally, changing the setpoint for B over time allows creating interesting trajectories like a spiral or an 8-figure, as shown from simulation in Fig. 15. uch trajectories were reproduced with Roball-2, but with less precision because of the limitations of its onboard sensors. However, such sensors are sufficient for generating interesting trajectories that will keep children engaged in interacting with the robot. Fig. 13. Radius of curvature in relation to the angle of the counterweight, for different rotational velocities of the sphere. Fig. 15. piral (top) and 8-figure (bottom) trajectories with = 3 rad/s over a 2m 2m area. The assigned setpoints for B over time are shown on the left, and on the right top views of the trajectories made by the robot are shown.

6 IV. CONCLUION Manufacturing custom spherical shelves is expensive, and being capable of anticipating the influence of the robotic components weights and size before fabricating the robot would be quite helpful. pherical robots have complex dynamics, and being able to simulate their behavior would be quite valuable in the design process. In this paper, we have presented non-linear and linear models of longitudinal and lateral motions of a spherical rolling robot that applies a torque at the rolling axis and can control the position of a counterweight for steering. Experimental trials done using Roball-2 confirm the validity of the models, considering the resolution of its onboard sensors. Future work involves the design of new prototypes of Roball of different sizes and weights, making the shell as robust and light as possible, and to use these prototypes to study infant-robot interaction [13]. As of now, Roball s prototypes were designed without any use of models to set its size, weight, radius of the flat surface of its side, etc. The models described in this paper will be useful in helping evaluate the motion capabilities of different versions of Roball before going into fabrication. [12] J. uomela and T. Ylikorpi, Ball-shaped robots: An historical overview and recent development at TKK, Field and ervice Robots, TAR 25, pp , 26. [13] J.-F. Laplante, F. Michaud, H. Larouche, A. Duquette,. Caron, D. Létourneau, P. Masson, Autonomous spherical mobile robot to study child development, IEEE Trans. on ystems, Man, and Cybernetics, Part A, vol. 35, no. 4, pp , 25. [14] J.-F. Laplante, Étude de la dynamique d un robot sphérique et de son effet sur l attention et la mobilité de jeunes enfants, Masters thesis, Department of Mechanical Engineering, Université de herbrooke, herbrooke, Québec Canada, 24. [15] G. D. Woods, imulating mechanical systems in imulink with immechanics, The MathWorks inc., 23. ACKNOWLEDGMENT The authors want to acknowledge the contributions of erge Caron and Dominic Létourneau in building Roball s second prototype. pecial thanks to Michel Lauria for his helpful comments in reviewing this paper. REFERENCE [1]. Halme, T. chönberg, and Y. Wang, Motion control of a spherical mobile robot, in Proc. Int. Workshop on Advanced Motion Control, Japan, [2] K. Husay, Instrumentation of a pherical Mobile Robot, Master s Thesis, Department of Engineering Cybernetics, Trondheim , 23. [3] A. Koshiyama and K. Yamafuji, Design and control of an alldirection steering type mobile robot, The International Journal of Robotics Research, vol. 12, no. 5, pp , [4] A. Bicchi, A. Balluchi, D. Prattichizzo, A. Gorelli, Introducing the phericle: An experimental testbed for research and teaching in nonholonomy, in Proc. IEEE Int. Conf. on Robotics and Automation, [5]. Bhattacharya and. K. Agrawal, pherical rolling robot: A design and motion planning studies, IEEE Transactions on Robotics and Automation, vol. 16, no. 6, pp , 2. [6] R. Mukherjee and M. Minor, A simple motion planner for a spherical mobile robot,, in Proc. IEEE/AME Int. Conf. on Advanced Intelligent Mechatronics, pp , [7] R. Mukherjee and T. Das, Feedback stabilization of a spherical mobile robot, Proc. IEEE/RJ Int. Conf. on Intelligent Robots and ystems, pp , 22. [8] A. Javadi and P. Mojabi, Introducing August: A novel strategy for an omnidirectional spherical rolling robot, in Proc. IEEE Int. Conf. on Robotics and Automation, pp , 22. [9] B. Chemel, E. Mustcheler, and H. chempf, Cyclops: Miniature robotic reconnaissance system, in Proc. IEEE Int. Conf. on Robotics and Automation, [1] F. Michaud and. Caron, Roball, the rolling robot, Autonomous Robots, vol. 2, no. 12, pp , 22. [11] F. Michaud and. Caron, Roball the rolling robot (Patent style), U.. Patent #6,227,933, May 8, 21.