J. Automation & Systems Engineering 5-2 (2011

J. Automation & Systems Engineering 5-2 (2011): 96-111 KINEMATIC ANALYSIS OF WHEELED MOBLIE ROOBT B B V L Deepak , Dayal R Parhi

JASE Journal of Automation & Systems Engineering

Abstract- This paper concerned with the kinematic analysis of a wheeled mobile robot (WMR). A wheeled mobile robot here considered as a planer rigid body that rides on an arbitrary number of wheels. In this paper it is shown that, for a large class of possible configurations of wheels, five types of categories can be done namely i) fixed standard wheels, ii) steerable standard wheels, iii) castor wheels, iv) Swedish wheels, and v) spherical wheels. These wheels are characterized by generic structures of the model equations. Based on the geometrical constraints of these wheels a kinematical model has been proposed and the degree of mobility, steerability and maneuverability are studied. Finally this analysis is applied to various three wheeled mobile robots. Examples are presented to illustrate the models Keywords: wheeled mobile robot, kinematic analysis, degree of maneuverability

1.

INTRODUCTION

Kinematics is the study of the mechanical systems behavior without considering the forces that affect the motion. In mobile robotics, it is necessary to familiar with the mechanical behavior of the robot both in order to design suitable mobile robots for tasks and to understand how to create control software for an instance of mobile robot hardware. A mobile robot is a self-operating machine that can move with respect to its free space. By understanding the wheel constraints placed on the robots mobility it is easier to understand the mobile robot motion. It is difficult to measure a mobile robot’s position instantaneously by direct way. It is an extremely challenging task to express the exact mobile robot’s position while it is in motion. The requirement for a mobile robot motion control is shown in Fig.1.

Fig.1 Requirements for motion control In literature [1] wheel architecture has been developed for the holonomic mobile platform in order to provide omni-directional motions by three individually driven and steered wheels. It has been investigated the kinematics of a mobile robot with the proposed double-wheel-type active caster [2] which is developed as a distributed actuation module and can endow objects with mobility on the planar workspace. For a mobile robot equipped with N Swedish wheels; kinematic modeling, singularity analysis, and motion control [3] has been outlined. In [4] an interactive tool has been described for solving mobile robot motion problems by understanding several well-known algorithms and techniques. A new concept has been introduced in the control and kinematic design [5] of mulit degree of freedom (MDOF) mobile robots. To control wheel slippage compliant * Corresponding author: B B V L Deepak Department of mechanical engineering, ,National Institute of Technology- Rourkela, E-mail: [email protected] ,E-mail: [email protected] Copyright © JASE 2011 on-line: jase.esrgroups.org

J. Automation & Systems Engineering 5-2 (2011): 96-111

linkage is provided compliance between the drive wheels or drive axles of a vehicle. A University of Minnesota’s Scout [6] is a small cylindrical robot capable of rolling and jumping for motion prediction and simulation has been modeled. Kinematic analysis has been performed for a subassembly [7], in which attachments of five legs in a Stewart platform are collinear on one side and coplanar on the other, paying particular attention to the problem of moving the aforementioned attachments without altering the singularity locus of the platform. In [8] it has been modeled that the wheels as a torus and proposed the use of a passive joint allowing a lateral degree of freedom for kinematic analysis of a wheeled mobile robot (WMR) moving on uneven terrain. A cross-coupling control algorithm [11] has been developed that guarantees a zero steady-state orientation error (assuming no slippage) and a stability analysis of the control system is presented. In [9] the structure of the kinematic and dynamic models has been analyzed for various wheeled mobile robots. A wheeled mobile robot has been modeled in [10] as a planar rigid body that rides on an arbitrary number of wheels. And they developed a relationship between the rigid body motion of the robot and the steering and drive rates of wheels. In this paper we consider a general wheel mobile robot (WMR), with three wheels of various types. Our aim is to point out the kinematic models for various WMRs. It has been shown that the variety of possible robot constructions and wheel configurations by the concepts of degree of mobility and of degree of steeribility. The set of WMR can be partitioned in to five configurations and this analysis is carried out in later sections. 2.

KINEMATICS OF VARIOUS WHEELED MOBILE ROBOTS:

2.1. Robot position representation: A wheeled mobile robot is a wheeled vehicle which is capable of an autonomous motion (without external human driver) because it is equipped with motors that are driven by an embarked computer for its motion. Throughout the analysis it has been considered that the robot as a rigid body on wheels and moves on a horizontal plane. The total dimensionality of this robot chassis on the plane is three, two for position in the plane and one for orientation along the vertical axis, which is orthogonal to the plane.

Fig.2 The global reference plane and the robot local reference frame Fig.2. represents relationship between the global reference frame of the plane and the local reference frame of the robot in order to find out the position of the robot on the plane. on the plane as the global Let us consider an arbitrary inertial frame O: reference frame and the robot’s local reference frame. To specify the position of the robot, choose a reference point P on the robot chassis as its position. The position of

97

B B V L Deepak & Dayal R Parhi: KINEMATIC ANALYSIS OF WHEELED MOBLIE ROOBT

P in the global reference frame is specified by coordinates x and y, and the angular difference between the global and local reference frames is given by θ. Therefore the robot position: (1) And mapping is accomplished using the orthogonal rotation matrix: (2) From equation (2) we know that we can compute the robot’s motion in the global reference frame from motion in its local reference frame: (3) 2.2. Wheel kinematic constraints: We can observe two constraints for every wheel type while a wheeled robot is in motion. The first constraint enforces the concept of rolling contact as represented in Fig.3.The second constraint enforces the concept of no lateral slippage, that the wheel must not slide orthogonal to the wheel plane as shown in Fig.4. The initial stage of a kinematic model of the robot is to express constraints on the motions of individual wheels. Thereby we can compute the movement of the entire robot by combining the motions of individual wheels.

Fig.3 Rolling motion Fig.4 Lateral slip Based on the geometrical constraints we can categorize the five basic wheel types: 1. Fixed standard wheel 2. Steered standard wheel 3. Castor wheel 4. Swedish wheel 5. Spherical wheels We assume that, the wheel always remains in vertical during the motion of a robot and there is no sliding at the single point of contact between the ground plane and the wheel. It means that the wheel is in motion under only pure rolling conditions and rotation about the vertical axis through the contact point. 2.2.1. Fixed standard wheel: There is no vertical axis of rotation or steering for the fixed standard wheel. It means the angle between the chassis and the wheel axis is fixed, therefore it is limited to move the robot back and forth along the wheel plane and rotation around its contact point with the ground plane. Fig.5 describes a fixed standard wheel and its position relative to the robot’s local reference frame. The position of the robot is then expressed in polar coordinates by distance l and angle α. β denotes the angle of the wheel plane relative to the robot chassis. Consider a wheel of radius r has its rotational position around its horizontal axle is a function of time t: φ (t). By the adequate amount of wheel spin in order to get pure rolling at the contact point, the wheel imposes that all movement along the direction of the wheel plane: (4)

98


The sliding constraint for this wheel enforces the wheel’s motion normal to the wheel plane must be zero:

Fig.5 Fixed standard wheel and its parameters 2.2.2. Steered standard wheel: The only difference between the fixed standard wheel and the steered standard wheel is that there is an additional degree of freedom: the wheel can rotate around a vertical axis passing through the ground contact point and the center of the wheel. i.e., the angular orientation of the wheel to the robot chassis is no longer a single fixed value , it may varies as a function of time: .

99


Fig.6 Steered standard wheel and its parameters The rolling and sliding constraints for the steered standard wheel shown in Fig.6: Along the wheel plane: (6) Orthogonal to the wheel plane: (7) 2.2.3. Castor wheel: Unlike the steered standard wheel, the vertical axis of rotation does not pass through the ground contact point in a castor wheel. These wheels are able to steer around a vertical axis. To specify the castor wheel’s position it is required an additional parameter which is a fixed length of rigid rod ‘d’ connected to wheel as shown in Fig.7. For the caster wheel, the rolling constraint is similar to the fixed standard wheel because the offset axis plays no role during motion aligned with the wheel plane: (8) Because of the offset ground contact point relative to A, the constraint that there is some lateral movement would be presented. Instead, the constraint is much like a rolling constraint, in that appropriate rotation of the vertical axis must take place: (9)

100


Fig.7 Castor wheel and its parameters 2.2.4. Swedish wheel Like the castor wheel, Swedish wheels are able to move omnidirectionally although these wheels have no vertical axis of rotation. This motion is possible by adding a degree of freedom to the fixed standard wheel. Swedish wheels configured like a fixed standard wheel with rollers attached to the wheel circumference with antiparallel axes to the main axis of the fixed wheel component. The exact angle γ between the main axis of the wheel and the roller axes may vary, as shown in Fig.8. The obtained motion constraint is similar to the fixed standard wheel formula is modified by adding γ such that the desired direction along the rolling constraint holds is along this zero component instead along the wheel plane. 10) Because of the free rotation of the small rollers, the motion is not constrained in orthogonal direction. 11)

Fig.8 Swedish wheel and its parameters the zero component of velocity is For a Swedish 90-degree wheel in which following with the wheel plane. So equation (10) reduced to equation (4) for the fixed standard wheel rolling constraint. But there is no sliding constraint orthogonal to the wheel plane because of the rollers. At the other extreme that are parallel to the main wheel axis of rotation. For the fixed standard wheel sliding constraint, equation (5).

the rollers have axes of rotation in equation (10) the result is

101


2.2.5. Spherical wheel The final wheel type is a spherical wheel as shown in Fig.9 which has no direct constraints on motion. In this configuration there are no principal axes of rotation, and therefore no rolling or sliding constraints may exist. Therefore equation (12) simply represents the roll rate of the sphere in the direction of motion vA of point A.

Fig.9 Spherical wheel and its parameters 3.

ROBOT KINEMATIC CONSTRAINTS: We now consider a wheeled mobile robot having N wheels of the above five described categories. We use the five following subscripts to identify quantities relative to these 5 classes: f for fixed wheels, s for steerable standard wheels, c for castor wheels, sw for Swedish wheels and sp for spherical wheel. The numbers of wheels of each type are denoted Nf, Ns, Nc, Nsw, and Nsp with Nf+ Ns+ Nc+ Nsw+ Nsp=N. The following vectors of coordinates describe the configuration of the robot. • for the five types • Angular coordinates: of wheels respectively. for the • Rotational coordinates: rotation angles of the wheels around their horizontal axis of rotation. Now it is easier to express the rolling constraints of all wheels into a unique expression as represented by equation (14). (14)

102


denotes a matrix for all wheels to their motions along their individual Where wheel planes, and J2 is a constant diagonal matrix NXN of all standard wheels radii.

=

(15)

where J1f , J1s , J1c , J1sw , and J1sp are the matrices of (Nf x 3), ( Ns x 3), (Nc x 3), (Nsw x 3) and (NspX3), whose forms are derived from the constraints (4), (6), (8), (10) and (12). J2 is a constant (N x N) matrix whose diagonal entries are the radii of the wheels, except for the radii of the Swedish wheels which are multiplied by cosγ. In the similar way we can formulate for the sliding constraints by combining all wheels into a single expression: 16)

Where

=

and

(17)

4.

MOBILE ROBOT MANEUVERABILITY The ability to move in the environment is the kinematic mobility of a robot chassis. The basic constraint limiting mobility is that every wheel must satisfy its sliding constraint. Besides the instantaneous kinematic motion, a mobile robot is able to further manipulate its position, over time, by steering steerable wheels. The overall maneuverability of a robot is thus a combination of the mobility available based on the kinematic sliding constraints of the standard wheels, plus the additional freedom contributed by steering and spinning the steerable standard wheels. 4.1. Degree of mobility: It is obvious from the wheel kinematic constraints described above that the castor wheel, Swedish wheel, and spherical wheel impose no kinematic constraints on the robot chassis, since these wheels can freely move in all of these cases during its internal wheel degrees of freedom. Therefore it is required to consider only fixed standard wheels and steerable standard wheels impact on robot chassis kinematics when computing the robot’s kinematic constraints. Consider now the (Nf + NS) wheels of fixed and steered standard wheels. To avoid any lateral slip the motion vector has to satisfy the following constraints: (18) (19) And

= Mathematically it represents must belong to the null space of the projection matrix. Null space of is the space N such that for any vector n in N (20) By using the concept of a robot’s instantaneous center of rotation (ICR) the kinematic constraints (equations (18) and (19)) can be explained geometrically. According to this at any given instant, wheel motion must be zero along the zero motion line. It means when a 103


robot takes turn, the wheel must be moving along some circle of radius such that the center of that circle is located on the zero motion line. This center point may lie anywhere along the zero motion line and is called the instantaneous center of rotation. When the wheel moves in a straight line its ICR is at infinity. This ICR geometric construction illustrates how robot mobility depends on the number of constraints on the robot’s motion but not the number of wheels. A robot like an Ackerman vehicle in Fig.10 (a) with several wheels, but it must have a single ICR. Because it’s zero motion lines meet at a single point. By placing the ICR at this meet point there is a single solution for robot motion. In Fig.12 (b), the bicycle like robot has two wheels W1, and W2. Each wheel gives a constraint, or a zero motion line. The solution for the ICR is only a single point by taking the two constraints together.

Fig.10 (a) Four wheel with car like Ackerman steering (b) bicycle Therefore kinematics of robot chassis is a function of independent constraints arising from all standard wheels. The mathematical significance of independence is corresponded to the rank of a matrix. Equation (13) represents all sliding constraints imposed by the is the number of independent wheels of the mobile robot. Therefore constraints. In practice, a wheeled mobile robot will have zero or more fixed standard wheels and zero or more steerable standard wheels. We can therefore identify the possible range of rank values for any robot: 0 . Now we can define a robot’s degree of mobility : (20) 4.1.1. Examples: . It means the robot 1. The maximum possible condition when rank of is completely constrained in all directions and is, therefore, degenerate since motion in the plane is totally impossible. Fig.11 represents how the robot is completely constrained by considering three fixed wheels.

Fig.11 robot cannot move anywhere (no ICR) and =2: 2. For a differential-drive chassis, rank A differential-drive robot shown in Fig.14 can control the change rates in both orientation and its Forward/reverse speed by manipulating wheel velocities and its ICR is constrained to lie on the infinite line extending from its wheels axles.

104


Fig.12 differential-drive robot and =1: 3. For a bicycle chassis as shown in Fig.13, rank There is one less degree of mobility compared to differential-drive robot. A bicycle only has control over its forward/reverse speed by direct manipulation of wheel velocities. Only by steering the bicycle can change its ICR.

Fig.13 Fixed arc motion, bicycle chassis . 4. The other extreme condition when rank of This is only possible if there are zero independent kinematic constraints in . In this case there are neither fixed nor steerable standard wheels attached to the robot frame. Fig.14 represents the robot has three castor wheels and ICR can be located at any position.

Fig.14 Robot having fully free motion 4.2. Degree of steerability: A wheeled robot can be steered independently by equipped with the number of steered standard wheels to it. This result is indirect since the robot must move for the change in steering angle of the steerable standard wheel. Therefore the degree of steerability can be obtained by: rank (21) Greater maneuverability means more degrees of steering freedom. It can be obtained by increasing the rank of . The range of is ince includes , a steerable standard wheel can both decrease mobility and increase steerability: • Its particular orientation at any instant imposes a kinematic constraint. • Its ability to change that orientation can lead to additional trajectories. 4.2.1. Examples: 1. For = 0: The robot has no steerable standard wheels, Ns = 0. 2. For =1: The most common case when a robot consists of one or more steerable standard wheels. For an ordinary vehicle (Fig.15) with Ns = 2 and Nf = 2, the fixed wheels share a common axle and thus rank of C1f = 1. The fixed wheels and one of the steerable wheels constrain the ICR. The second steerable wheel cannot impose any independent kinematic constraint and thus rank = 1. In this case, = 1and = 1.

105


Fig.15 Robot with two fixed & two steerable standard wheels 3. For = 2: Only possible in robots with no fixed standard wheels: Nf = 0. The pseudobicycle as shown in Fig.16 with two separate steerable standard wheels is an example for this case. Its ICR can be placed anywhere on the ground plane, which is also the meaning of = 2.

Fig.16 Pseudobicycle with two separate steerable standard wheels 4.3. Maneuverability of three wheeled Mobile Robot: The overall Degrees of Freedom (DOF) that a robot can manipulate is called the degree Thus the maneuverability comprises with the degrees of freedom of maneuverability that the robot changes its position directly through wheel velocity and the degrees of freedom that it indirectly manipulates by changing the steering configuration and moving. (22) 1. Omni directional: Fig.17 represents three wheeled omnidirectional mobile robot having three wheels other than fixed and steered standard wheels. For this type of robot and is zero since it has no steerable standard wheels and fixed standard wheels.

Fig.17 Three wheeled omnidirectional mobile robot 3 This results the degree of mobility and the degree of steerability 0 The degree of maneuverability =3 2. Differential: Fig.18 represents three wheeled differential mobile robot having two wheels are fixed and other is except fixed and steered standard wheels. For this type of robot is zero and is one since it has no steerable standard wheels.

Fig.18 Differential mobile robot 2 This results the degree of mobility and the degree of steerability 0 106


The degree of maneuverability =2 3. Omni-Steer: Fig.19 represents three wheeled omni-steer mobile robot having two wheels are other than fixed and steered standard wheels and one steered standard wheel. For this type of robot is one and is one since it has one steerable standard wheel.

Fig.19 Three wheeled omni-steer mobile robot 2 This results the degree of mobility and the degree of steerability 1 The degree of maneuverability =3 4. Tricycle: Fig.20 represents Tricycle mobile robot having two fixed wheels and one steered standard wheel. For this type of robot is one and is two since it has one steerable standard wheel.

Fig.20 Tricycle mobile robot 1 This results the degree of mobility and the degree of steerability 1 The degree of maneuverability =1 5. Two steer: Fig.21 represents two steer three wheeled mobile robot having two steered standard wheels and one omnidirectional wheel. For this type of robot is two and is two since it has two steerable standard wheels.

Fig.21 Two steer mobile robot 1 This results the degree of mobility and the degree of steerability 1 The degree of maneuverability =1 5. AN OMNIDIRECTIONAL THREE WHEELED MOBILE ROBOT EXAMPLE: Consider the omniwheel robot shown in Fig.22 having three Swedish wheels. If the angle between the main wheel plane and the axis of rotation of the small circumferential rollers =0, it means the robot has three Swedish 90-degree wheels, arranged radially symmetrically, with the rollers perpendicular to each main wheel. Assume that the distance between each wheel and local reference frame is l.

107


Fig.22 Three wheel omnidrive robot Rolling constraint for a Swedish wheel: (21) If Equation (14) becomes: (22) Suppose all wheels are tangent to the robot circular body as shown in Fig.23 implies then equation (22) becomes: (23)

Fig.23 Local reference frame plus detailed parameters for wheel 1. To map motion along the axes of the global reference frame to the robot’s local reference frame mapping matrix R(θ) is used. Here θ is the angular difference between the global and local reference frame. (23) From equation (13), the rolling constraints of all wheels can be expressed: (24) And = Since the robot has no steered standard wheels equation (16) becomes:

can be taken as

(25) . Therefore (26) (27)

108


The

value

of

for

three

wheels

easily

computed:

=

= J2 is the diagonal matrix of radii of the three wheels. Let us take r1=1, r2=1 and r3=1:

= Consider an omnidrive chassis with l=1, and θ=00 i.e., the robots local reference frame and global reference are aligned. If wheels 1, 2, and 3 spin at speeds =1,

From the consideration And Therefore the motion of the robot from the equation (24): (28)

Degree of maneuverability: Orthogonal motion of omnidrive robot: (29) 109


For no lateral slip: (30) And = Since this robot has three Swedish wheels, and no standard wheels. Therefore the is zero. From this we can determine degree of mobility: = 3-0= 3 And degree of steerability is =0 = 3+0=3 6.

CONCLUSION: In this paper, the kinematics modeling for various wheeled mobile robots is proposed based on a geometric approach. All WMR are classified into five configurations according to the mobility restriction imposed by the kinematic constraints. It has been analyzed and derived a robot kinematics model for an N wheeled mobile. The derived kinematic model is adequate to describe the global motion of the robot. Obtained results from the present research can be applied to areas in the automotive industry, especially to all wheel drive electric vehicles. As a future work, it is required to develop the control algorithms and the motion experiments for real application. REFERENCES: [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

110

Dong-Sung Kim, Wook Hyun Kwon, and Hong Sung Park, “Geometric Kinematics and Applications of a Mobile Robot”, International Journal of Control, Automation, and Systems Vol. 1, No. 3, September 2003, pp 376-384. Jae Hoon Lee, Bong Keun Kim, Tamio Tanikawa and Kohtaro Ohba, “Kinematic Analysis on Omni-directional Mobile Robot with Double-wheel-type Active Casters”, International Conference on Control, Automation and Systems, COEX, Seoul, Korea, 2007, pp 1217-1221. Giovanni Indiveri, “Swedish Wheeled Omnidirectional Mobile Robots: Kinematics Analysis and Control”, IEEE transactions on robotics, VOL. 25, NO. 1, February 2009, pp 164-171. J.L. Guzm´ana, M. Berenguela, F. Rodr´ıgueza, and S. Dormidob, “An interactive tool for mobile robot motion planning”, Robotics and Autonomous Systems 56, 2008, pp 396–409. Johann Borenstein, “Control and Kinematic Design of Multi-Degree-of- Freedom Mobile Robots with Compliant Linkage”, IEEE transactions on robotics and automation, vol. 1, February 1995, pp 21-35. Sascha A. Stoeter and Nikolaos Papanikolopoulos, “Kinematic Motion Model for Jumping Scout Robots”, IEEE transactions on robotics, vol. 22, no. 2, April 2006, pp 398-403. Júlia Borràs and Federico Thomas, “Kinematics of Line-Plane Subassemblies in Stewart Platforms”, IEEE International Conference on Robotics and Automation, Kobe, Japan, May 2009, pp 4094-4099 Nilanjan Chakraborty and Ashitava Ghosal, “Kinematics of wheeled mobile robots on uneven terrain”, Mechanism and Machine Theory 39, 2004, pp 1273–1287.


Guy Campion, Georges Bastin, and Brigitte D’ AndrCa-Novel, “Structural Properties and Classification of Kinematic and Dynamic Models of Wheeled Mobile Robots”, IEEE transactions on robotics and automation, vol. 12, no. 1, February 1996, pp 47-62. [10] J. C. Alexandery and J. H. Maddocks, “On the Kinematics of Wheeled Mobile Robots”, University Of Maryland, October, 1988, pp 1-20. [11] Johann Borenstein+ and Yoram Koren, “Motion Control Analysis Of A Mobile Robot”, Transactions of ASME, Journal of Dynamics, Measurement and Control, Vol. 109, No. 2, November 1986, pp 73-79. [9]

111