10 a review of models and structures for wheeled mobile robots four case studies

The 15th International Conference on Advanced Robotics Tallinn University of Technology Tallinn, Estonia, June 20-23, 2011 A Review of Models and Structures for Wheeled Mobile Robots: Four Case Studies Ramiro Vel´azquez and Aim´e Lay-Ekuakille Abstract— This paper reviews the mathematical models of four commonly encountered designs for wheeled mobile robots (WMR) These designs belong to two generic classes of wheeled robot structures: differential-drive and omnimobile First, the two wheel differential-drive model is presented to show how zero turning radius is achieved with only bidirectional movement Three particular designs are addressed: the popular two-active-fixed wheels and one-passive-caster wheel, a simple belt-drive, and sprocket-chain system Next, the model for omnimoble robots with Swedish wheels is presented to illustrate holonomic omnidirectional motion All four models are based on physical parameters easily measured and are useful to understand the internal dynamics of these WMR and to accurately visualize their motion in 2D environments They can be therefore used as a practical reference to predict the accessibility of physical prototypes to selected places and to test different algorithms for control, path planning, guidance, and obstacle avoidance their orientations must point to the same direction (s=1) Therefore, mobility is restricted to a two-dimensional plane (m=2) An example is the synchronous drive WMR in [6] • Type (1,1) robots have one or several fixed wheels on a common axle and also one or several steering wheels, with two conditions for the steering wheels: their centers must not be located on the common axle of the fixed wheels and their orientations must be coordinated (s=1) Mobility is restricted to a onedimensional plane determined by the orientation angle of the steering wheel (m=1) Examples of this type are the tricycle, the bicycle, and the car-like WMR • Type (1,2) robots have no fixed wheels, but at least two steering wheels If there are more than two steering wheels, then their orientation must be coordinated in two groups (s=2) Mobility is restricted to a onedimensional plane (m=1) determined by the orientation angles of the two steering wheels This paper particularly address type (3,0) and (2,0) robots Taking as example our own prototypes (and some practical lessons learned from their implementation), we derive the mathematical models of four commonly encountered designs for these two types of WMR The rest of the paper is organized as follows: in Section 2, the popular two wheel differential-drive model is obtained using the general two-active-fixed wheels and onepassive-caster wheel structure Next, two other differentialdrive designs are presented to illustrate some other efficient locomotion systems: a simple belt-drive system which shows how frictional forces transfer torque to generate motion and a sprocket-chain system which offers another method for transferring motion when frictional forces are insufficient to transfer power In Section 3, the omnimobile robot with Swedish wheels is analyzed The resulting model shows how holonomic omnidirectional motion is achieved Finally, the conclusion summarizes the paper main concepts I I NTRODUCTION Understanding how wheeled mobile robots (WMR) move in response to input commands is essential for feedback control design and many navigation tasks such as path planning, guidance, and obstacle avoidance Campion and Chung classified in [1] the mobility of WMR into five generic structures corresponding to a pair of indices (m, s): mobility degree m and steerability degree s The first one refers to the number of degrees of freedom the WMR could have instantaneously from its current position without steering any of its wheels while the second refers to the number of steering wheels that can be oriented independently in order to steer the WMR These five classes are: • Type (3,0) robots or omnidirectional robots have no steering wheels (s=0) and are equipped only with Swedish or active caster wheels They have full mobility in the plane (m=3), which means that they are able to move in any direction without any reorientation Representative examples of such robots are [2] and [3] • Type (2,0) robots have no steering wheels (s=0) but either one or several fixed wheels with a common axle The common axle restricts mobility to a twodimensional plane (m=2) Examples of type (2,0) robots are [4] and [5] • Type (2,1) robots have no fixed wheels and at least one steering wheel If there is more than one steering wheel, II D ESIGNS AND P ROTOTYPES Let us start addressing type (2,0) robots There are many design alternatives; however, the two-wheel differential-drive robot is by far the most popular design Let us consider our prototype IVWAN (Fig 1(a)) Its mechanical structure is based on a differential-drive configuration consisting of two independently controlled frontactive wheels and one-rear-caster wheel (Fig 1(b)) Active wheels are driven by two high-power DC motors which allow IVWAN to achieve a maximum speed of 20 km/hr R Vel´azquez is with the Mechatronics and Control Systems Lab (MCS), Universidad Panamericana, 20290, Aguascalientes, Mexico Contact: rvelazquez@ags.up.mx A Lay-Ekuakille is with the Department of Innovation Engineering, Universit`a del Salento, 73100, Lecce, Italy Contact: aime.lay.ekuakille@unisalento.it 978-1-4577-1159-6/11/$26.00 ©2011 IEEE 524 (a) (b) (c) Fig Type (2,0) WMR IVWAN (Intelligent Vehicle With Autonomous Navigation): (a) prototype and (b) its differential-drive structure Two front wheels each driven by its own motor A third wheel is placed in the rear to passively roll along while preventing the robot from falling over The wheels exhibit three speeds: u, u ¯, and ω (c) Free-body diagram The first subscript stands for front f and caster c wheel while the second subscript stands for right r and left l wheel Kinematics of point G is related to u and ω by eq (3): IVWAN exhibits both manual and autonomous operation: it can be tele-operated or self-guided by a color camera and an array of ultrasonic sensors that allow the machine to detect and follow visual patterns and negotiate obstacles, respectively [7] Fig 1(c) shows a schematic representation of the differential-drive structure Here, B represents the center of the axis connecting both traction wheels; G represents the vehicle’s center of mass and for simplicity, it is considered as the point to control in position (x, y) and orientation (ϕ) Resultant forces and momentum in the structure can be expressed by eq (1):    Fx = m(u˙ − u¯ω) = Ff rx + Ff lx + Fcx + FGx Fy = m(u ¯˙ + uω) = Ff ry + Ff ly + Fcy + FGy Mz d (Ff rx − Ff lx ) − b(Ff ry + Ff ly ) + (1) +(c − b)Fcy + τG where m is the vehicle’s total mass, I is the moment of inertia around point G, and u, u¯ and ω are the robot’s linear, transverse sliding, and angular speeds, respectively (Fig 1(b)) Speed u ¯ can be reasonable neglected assuming that the wheels not slip during motion Concerning u and ω, they can further be defined by eq (2): u = ω = = ucosϕ − bωsinϕ = usinϕ + bωcosϕ ϕ˙ = ω (3) As aforementioned, traction wheels are powered by DC motors These can be modeled by eq (4): ka (Er − kb ωr ) τr = Ra ka (El − kb ωl ) (4) τl = Ra where τr and τr are the torques developed by the motors on the right and left wheels upon input DC voltages Er and El respectively, ka and kb are the motor’s torque and electromotive force constants, and Ra is the motor’s electric resistance Inductive voltages have been neglected Equations describing the wheel-motor system can be simply written as shown in eq (5): = I ω˙ = [r(ωr + ωl ) + (ur + ul )] [r(ωr − ωl ) + (ur − ul )] d x˙ y˙ Ie ω˙r + De ωr = τr − Ff rx rˆ Ie ω˙l + De ωl = τl − Ff lx rˆ (5) where Ie and De are the moment of inertia and the coefficient of viscous friction of the wheel-motor system, respectively and rˆ is the nominal radius of the traction wheel tires Using and combining eqs (1) to (5), the differentialdrive model can be summarized by eq (6): ⎤ ⎡ ⎡ ⎤ ⎡ ucosϕ − bωsinϕ x˙ ⎢ y˙ ⎥ ⎢ usinϕ + bωcosϕ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥+⎢0 ⎢ϕ˙ ⎥ = ⎢ ω ⎥ ⎢ ⎢ ⎥ ⎢ a3 a ⎣ u˙ ⎦ ⎣ a rˆrω − a4 u ⎦ ⎣ a2r 1 −2 aa23 rˆruω − aa42 d2 ω ω˙ (2) where r is the traction wheel radius, d is the distance between the traction wheels (see Fig 1(c)), ωr , and ωl are the angular speeds of the right and left wheels respectively, and ur and ul are the linear speeds of the right and left wheels respectively with inputs: 525 Eu = Eω = Er + El Er − El 0 0 2rd a2 ⎤ ⎥ ⎥ Eu ⎥ ⎥ Eω (6) ⎦ (a) Fig (b) (a) Block diagram reference for differential-drive robots (b) Summary of motion upon voltages Er and El and constants: Ra a1 = (mˆ r r + 2Ie ) [V · s2 ] ka Ra a2 = [Ie d2 + 2ˆ rr(I + mb2 )] [V · m2 · s2 ] ka Ra mb [V · s2 /m] a3 = ka Ra ka kb ( + De ) [V · s/rad] a4 = ka Ra Note that eq (6) relates the robot’s motion to the motors’ input voltages The block diagram model for differentialdrive robots is shown in fig 2(a) This diagram identifies the electronics, DC motors, and the vehicle’s kinematics Fig 2(b) summarizes how differential-drive robots are controlled by the input voltages Er and El When both voltages are equal, the two driving wheels turn at the same angular speed and in the same direction, which causes a translation movement If one voltage is set to zero, one of the wheels turns while the other remains motionless, then the robot describes a circle centered on the motionless wheel If both voltages are equal in magnitude but opposite sign, the wheels turn at the same speed but in opposite direction which causes a rotation around the center of the axis connecting both wheels (point B) Note a zero turning radius in this case Numerical values of the parameters involved in eq (6) can be easily measured from an existent prototype and the specifications of the DC motors can be obtained from the manufacturer As illustrative example, consider all gain blocks of fig 2(a) as unity gains Fig 3(a) shows a computer simulation of a certain trajectory in the XY plane Fig 3(b) shows the driving signals supplied to the DC motors Note the correspondence with fig 2(b) There are other design alternatives for differential-drive robots, subsections A and B present two different approaches (a) (b) Fig (a) A simulated trajectory of the differential-drive robot and (b) the corresponding driving signals life They transmit power through frictional contacts They function best at moderate speeds (20 to 30 m/s) under static loads Their efficiencies drop slightly at low speeds and centrifugal effects limit their capacities at high speeds [8] Let us consider our prototype Enyo (Fig 4(a)) Its mechanical structure is based on a four wheel differential-drive configuration driven by a belt system Two active-frontwheels transfer rotating motion to the two passive-rearwheels through belts (Fig 4(b)) This motor-belt system allows Enyo to achieve a maximum speed of 30 km/hr Even though Enyo seems a car-like type (1,1) WMR, it is a type (2,0) WMR because none of its wheels are steerable Fig 4(c) shows a schematic representation of the belt-drive system Here, the belt is modeled as a spring with constant A Belt-drive Belts have long been used for the transfer of mechanical power Today’s flat belts are relatively light, inexpensive, tolerant of alignment errors, and ensure a long operating 526 (a) Fig (b) (c) Type (2,0) WMR Enyo: (a) prototype and (b) its belt-drive system (c) Schematic of the locomotion system TABLE I k The radii of the pulleys are r1 and r2 while their inertias are J1 and J2 , respectively As in the previous case, traction pulleys are power by DC motors The angular speed of the motor and hence, of the active pulley is θm and the angular speed of the passive pulley is θp Eqs (7) to (9) describe this system: Li J1 ăm J2 ăp ˙ = E − Ri − ke θm = kt i − bθ˙m − r1 (F1 − F2 ) = −bθ˙p + r2 (F1 − F2 ) S UMMARY OF THE PARAMETERS INVOLVED IN THE BELT- DRIVE MODEL Parameter R L ke kt J r b (7) (8) (9) Value 0.99 15.55 0.04 0.11 0.25 Unit Ω H V-s/rad N-m/A kg-m2 m N-m-s/rad where E is the input voltage to the motor, R and L are the motor’s electric resistance and inductance, kt and ke are the motor’s torque and electromotive force constants, respectively The motor-pulley friction is denoted by b while F1 and F2 are the forces exerted by the belt on the pulleys These forces can be further expressed as eq (10): F1 F2 = −F1 = = k(x1 − x2 ) k(x2 − x1 ) (10) Knowing that x1 = r1 θm and x2 = r2 θp , and further considering that pulleys are identical (J1 = J2 = J and r1 = r2 = r), eqs (8) and (9) become eqs (11) and (12): J ăm J ăp = = kt i bθ˙m − 2kr2 (θm − θp ) −bθ˙p + 2kr2 (θm − θp ) (11) (12) The motor-pulley-belt system can be summarized by state eq (13): ⎡ ⎤ ⎡ R ⎤ ⎡ ⎤ ⎡ ⎤ Fig Effect of varying the spring constant k on the belt-drive system i 0 − L − kLe i Plots for k=10, 50, 150, 1000 N/m L b 2kr 2kr ăm kt θ − − ⎢ ⎥ ⎢ J ⎥ ⎢ m⎥ ⎢ ⎥ J J J ⎢θ˙m ⎥ = ⎢ ⎢ ⎥ ⎢ ⎥ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢θm ⎥+⎢ ⎥ E ăp 2kr b 2kr ⎦ ⎣ θ˙ ⎦ ⎣ ⎦ under constant rotation, belts tend to creep Thus, these drives −J − J p J must be kept under substantial tension to function properly θp θ˙p 0 One possible physical implementation to increase k is the (13) Essentially, the goal of belt-drive systems is to transfer one shown in fig 4(b) A third pulley or V-belt pulley forces efficiently mechanical power between pulleys so that their the belt to increase it spring constant k angular speeds are the same (θm = θp ) Using Enyo’s parameters given in table 1, fig examines B Sprocket and chain drive Sprockets and chains offer another option for transferthe effect of varying the spring constant k on the pulleys’ angular speeds Note that, the higher the values of k, the best ring rotating motion between shafts when the friction of match between θm and θp In practice, this means that, even a drive-belt is insufficient to transfer power Contrary to 527 (a) Fig (b) (c) Type (2,0) WMR Connor: (a) prototype and (b) its sprocket and chain system (c) Schematic of the locomotion system belts, sprockets and chains transmit power through bearing forces while maintaining a fixed phase relationship between the input and output shafts The main drawback is the contact between the sprocket and chain: the contact can slip significantly as the chain rollers and sprocket teeth move in and out [9] Let us consider our prototype Connor (Fig 6(a)) Its mechanical structure is based on a sprocket and chain differential-drive configuration Using chains, one activefront shaft transfers rotating motion to two passive-rear shafts (Fig 6(b)) This drive allows Connor to achieve a maximum speed of 20 km/hr Its caterpillar type structure makes Connor a type (2,0) WMR as none of its sprockets are steerable Fig 6(c) shows a schematic representation of Connor’s sprocket and chain system As in a belt-drive, the chain can be modeled as a spring with constant k The radii of the sprockets are r1 , r2 , and r3 while their inertias are J1 , J2 , and J3 Angular speeds are θm , θp , and θq , respectively Considering again a DC motor with constants kt and ke as actuator, then the motor-sprocket-chain system can be summarized by eqs (14) to (17): Li J1 ăm J2 ăp J3 ăq = E Ri ke m (14) kt i − bθm − r1 k(2r1 θm − r2 θp − r3 θq )(15) = −bθ˙p + r2 k(2r2 θp − r3 θq − r1 θm ) (16) = r3 k(2r3 θq − r1 θm − r2 θp ) (17) = III O MNIMOBILE WMR Omnimobile WMR correspond to type (3,0) robots The main advantage of these WMR is that they exhibit holonomicity, i.e the ability to move in any direction without an orientation change The holonomicity and omnidirectional properties come from the use of Swedish wheels or active caster wheels Conventional wheels limit WMR motion as they exhibit DOF and a no side-slip condition (see Fig 1(b)) However, special wheels such as the Swedish actually ensure DOF Let us consider our prototype NG (Fig 7(a)) Its mechanical design is based on an equilateral triangle-like structure Three omniwheels are disposed at each one of its vertices and are directly coupled to DC motors (Fig 7(b)) NG can be tele-operated via an RF point-to-point connection or autonomously guided by an array of ultrasonic sensors It was conceived to participate in the 2009 Robocup middlesize soccer league Fig 7(c) shows a schematic representation of its omnimobile structure Here, five coordinate systems are defined: SI the inertial reference system with coordinates (X,Y), Sc the robot’s center of mass with coordinates (xc , yc ), and Si at each wheel with coordinates (xi , yi ), i=1,2,3 L is the distance between the coordinate systems SI and Si , ϕi is the wheel-i angle in the Sc system while θ is the robot’s orientation angle in the SI system The robot’s position in space is then defined by posture vector βI =[X Y θ]T The relationship between the robot’s velocity on the SI and Sc coordinate systems is defined by eq (18): In principle, all contact drives experience frictional losses The friction between all-metal drives without any lubricant is high enough to damage the drive On the other hand, beltdrives have little frictional losses and the sprocket and chain drive has none In eqs (15) and (16), b represents the friction between the sprocket-chain drive and the rolling surface As in a belt-drive, simulation of eqs (14) to (17) would illustrate the speed relationships between the three sprockets of different radii as well as the importance of having the chain under substantial tension Note that the third sprocket (J3 ) is meant for this purpose β˙I = R(θ)−1 β˙c with: (18) ⎡ ⎤ cos(θ) −sin(θ) R(θ) = ⎣sin(θ) cos(θ) 0⎦ 0 where R(θ) is an orthogonal rotation matrix from Sc to SI Similarly, velocity components on Sc are function of velocity components on the Si coordinate system This relationship is described by eq (19): 528 (a) Fig (b) (c) Type (3,0) WMR NG: (a) prototype, (b) equilateral triangle-like structure with Swedish wheels, and (c) schematic of the robot’s kinematics ⎡ ⎤ ⎡ x˙c cos(ϕi ) ⎣ y˙c ⎦ = ⎣sin(ϕi ) θ˙ ⎤⎡ ⎤ x˙i −sin(ϕi ) Lsin(ϕi ) cos(ϕi ) −Lcos(ϕi )⎦ ⎣ y˙i ⎦ (19) θ˙ Considering that ϕ1 = 90◦ , ϕ2 = 180◦, and ϕ3 = 330◦ in NG and applying the coordinate transformation described in eq (18), the robot’s inertial velocities can be expressed as a function of the wheels angular velocities ωi (eq (20)): ⎡ ⎡ ⎤ −3c X˙ ⎣ Y˙ ⎦ = r ⎣− s θ˙ 3L √1 3c − 3s √1 3s + 3c 3L ⎤⎡ ⎤ √1 ω1 3c + 3s √1 c⎦ ⎣ω2 ⎦ s − 3 ω3 3L (20) where c=cos(θ), s=sin(θ), and r is the omniwheels’ radius Simulation of eq (20) allows motion visualization of omniWMR like NG As example, consider L=0.3 m, r=0.05 m Fig shows a set of trajectories in the XY plane As it can be inferred, resulting motion is a combination of the angular velocities of the three wheels To move the omnidirectional robot in a straight line, the angular speeds of two wheels have to be set equal in magnitude but opposite in sign while the angular speed of the third wheel has to be set as zero To rotate the robot around any point, only one wheel has to be actuated To rotate it around its center of mass, all three angular speeds have to be equal in magnitude and sign Note that omni-WMR offer a wide range of motion possibilities which make them superior to differential-drive WMR in terms of dexterity and driving capabilities [10] Fig Simulated trajectories for omnidirectional robot NG Computer simulation of these models allows motion visualization and can be therefore exploited for the design of control algorithms, path planning, obstacle avoidance, etc R EFERENCES [1] G Campion and W Chung, “Wheeled robots”, Chapter 17 in: Handbook of Robotics (B Siciliano, O Khatib, eds.), Springer, pp 391-410, 2008 [2] L Huang, Y Lim, D Li and C Teoh, “Design and analysis of a fourwheel omnidirectional mobile robot”, Proc of International Conference on Autonomous Robots and Agents, pp 425-428, 2004 [3] M Wada and S Mori, “Holonomic and omnidirectional vehicle with conventional tires”, Proc of IEEE-ICRA, pp 3671-3676, 1996 [4] R Velazquez, A Camacho and B Romero, “Modeling, design and vision-based control of a low-cost electric power wheelchair prototype” Int Journal of Assistive Robotics and Systems, 10(3), pp 13-24, 2009 [5] E Papadopoulos and M Misailidis, “On differential drive robot odometry with application to path planning”, Proc of European Control Conference, pp 5492-5499, 2007 [6] D Fox, W Burgard and S Thrun, “Controlling synchro-drive robots with the dynamic window approach to collision avoidance”, Proc of IEEE-IROS, pp 1280-1287, 1996 [7] J.S Martnez, G Moran, B Romero, A Camacho, D Gutheim, J Varona, R Velazquez, “Multifunction all-terrain mobile robot IVWAN: design and first prototype”, Proc of Israeli Conf on Robotics, 2008 [8] N Sclater and N Chironis, Mechanisms and mechanical devices sourcebook, McGraw-Hill, New York, 2001 [9] J Uicker, G Pennock and J Shigley, Theory of Machines and Mechanisms, Oxford University Press, New York, 2003 [10] H Oliveira, A Sousa, A Paulo and P Costa, “Modeling and assessing of omnidirectional robots with three and four wheels”, Chapter 12 in: Contemporary Robotics (A Rodic ed.), InTech, pp 207-229, 2009 IV C ONCLUSION This paper aimed to present simple and reliable mathematical models for different designs of type (3,0) and (2,0) WMR It intends to be a practical reference for fast and easy understanding of the main equations governing these WMR In particular, this paper addressed differential-drive and omnimobile WMR For the first, the kinematics and dynamics of popular configurations such as the general two-activefixed wheels and one-passive-caster wheel, the belt-drive, and sprocket-chain were obtained For the second, the kinematics of WMR with Swedish wheels was presented All models were illustrated using physical prototypes 529 ... Huang, Y Lim, D Li and C Teoh, “Design and analysis of a fourwheel omnidirectional mobile robot”, Proc of International Conference on Autonomous Robots and Agents, pp 425-428, 2004 [3] M Wada and. .. This paper aimed to present simple and reliable mathematical models for different designs of type (3,0) and (2,0) WMR It intends to be a practical reference for fast and easy understanding of the... operation: it can be tele-operated or self-guided by a color camera and an array of ultrasonic sensors that allow the machine to detect and follow visual patterns and negotiate obstacles, respectively