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Kinematical Analysis of a Four Steered Wheeled Mobile Robot


Kinematical Analysis of a Four Steered Wheeled Mobile Robot
Michel Lauria, Isabelle Nadeau, Pierre Lepage, Yan Morin, Patrick Giguère, Fréderic Gagnon, Dominic Létourneau and Fran?ois Michaud
LABORIUS - Research Laboratory on Mobile Robotics and Intelligent Systems Department of Electrical Engineering and Computer Engineering

Université de Sherbrooke
Sherbrooke, Québec J1K 2R1, CANADA

{Michel.Lauria, Isabelle.Nadeau, Francois.Michaud}@USherbrooke.ca
Abstract - This paper presents the kinematical analysis of AZIMUT-2. The utilization of a new wheel concept called the AZIMUT wheel allowed us to create an innovative omnidirectional non-holonomic robot. Novelty of this wheel concept resides in the non-conventional positioning of the steering axis and the wheel axis. We propose a kinematical model based on the geometrical constraints of these wheels. The degree of mobility, steerability and maneuverability are studied. Additionally, we describe a special design implementation of the wheel mechanism to overcome an hyper-motorization issue inherent to this wheel geometrical properties. Finally, we describe AZIMUT-2’s two operational modes and its seven locomotion modes, along with a control algorithm based on the kinematical model of the robot.

I.

INTRODUCTION

Real world usage of mobile robotic platforms is limited primarily by the robot’s capability to move in and to perceive the world. Adopting a modular design approach, our objective is to design a multi-purpose robotic platform on which different locomotion modalities could be used. Over the last three years, we have worked on such platform (named AZIMUT), presenting two omnidirectional and modular research prototypes. They were designed to validate multimodal locomotion capabilities, providing simple reconfigurations (done through manual intervention or even autonomously by the robot) for various usages over different kinds of environments. AZIMUT’s concept aims at versatility in the locomotion modes, by combining wheels, tracks, legs, etc. into one platform. Designing a robot with flexible locomotion capabilities requires the integration of complex mechanical, electrical and computer components. Difficult design compromises (e.g., size versus weight versus energy versus heat dissipation versus responsiveness) must be addressed, with measurable impacts only after integrating all of the components together. It is an iterative process that requires more than just one prototype to come up with an appropriate solution to an application problem. Modularity at the structural, hardware and embedded

software levels, all considered concurrently during the design process, reveals to be key in the design of such sophisticated platform. It makes it possible to improve critical components while not having to reimplement the entire concept, facilitating continuous improvements of the platform over time. Modularity facilitates such reconfigurations, and at the same time reduces cost (allowing to reuse core elements in different configurations) and maintenance effort. Our interest in this paper is on AZIMUT’s wheeled configuration. It has been established by various researchers that the number, the nature and the disposition of the wheels determine the kinematical properties of a robot. As of now, five categories of wheels have been identified: the fixed standard wheel, the steered standard wheel, the castor Wheel, the Swedish wheel and the spherical wheel [1]. In this paper, we introduce a new category of wheel, named the AZIMUT wheel. In the following sections, we present and analyze the kinematical properties of this new kind of wheel. We also present design guidelines related to the implementation of this kind of wheel. II. PRESENTATION OF AZIMUT-1 AND AZIMUT-2

To conduct research activities on advanced kinematic and locomotion control of mobile robots with concrete results, real platforms are required and needed to be designed. With the goal of designing a robotic platform combining different locomotion mechanisms and intended to operate in home-like environments, AZIMUT’s first prototype (AZ1) [2], shown in Figure 1, was designed in 2002. AZ1 is symmetrical and is made of four independent articulations. Each articulated part combines a leg, a track and a wheel, and has 3 degrees of freedom (DOF) (propulsion, direction and rotation). The leg can rotate 360° around its attachment point and 180° to change its direction. Once an articulation is placed at the right position, the system is designed to keep it in position without consuming electrical energy. When the articulations are stretched, the robot can move by making the tracks rotate around the legs. As the articulations move upward toward the vertical, the point of

Fig. 1 AZIMUT-1

autonomy depending on the operating conditions. AZ1 demonstrated its omnidirectional and stair/slope handling capabilities. It was, however, only functional at 80% of its locomotion capabilities, which was still very well considering the complexity of the design and the constraints in time, budget and resources. While it was possible to come close to the desired specifications for the width, the height, the body clearance and the articulation blocking torque, the first prototype is heavier (63.5 kg instead of 54.4 kg) than expected. AZ1’s chassis had to be made using aluminum and steel parts without weight optimization, and the articulation lifting torque is lower than what was evaluated using the motor specifications (which revealed to not be completely accurate) and the mechanism designed. This explains why AZ1 could not lift itself up. It also adopts a distributed embedded design approach. Lowlevel control of the platform is done using distributed subsystems (for motor control, sensing, energy distribution, etc.), each equipped with its own microcontroller board (PIC, Infineon, Motorola, etc., determined according to the computing requirements). Communication between the microcontrollers is done using Control Area Network protocol (CAN 2.0B) on a 1 Mbps shared bus, and our own 92 bits frame protocol. Improved robustness and flexibility is therefore provided, since microcontroller boards can be easily switched or replaced in case of problems. In 2004, we initiated the design of AZIMUT’s second prototype (AZ2 – Fig. 3), with the goals of improving over the last design and increasing modularity of the structural components of the platform: ? Increased modularity is achieved by having detachable articulations. AZ2 can be configured to use wheels, legs or tracks, or combination of them. This led to the design of a wheel-motor (made with a brushless motor coupled with a

contact of the leg with the ground moves from the track to the rubber strip fixed outbound of the attachment axle of the articulation. This rubber strip creates a very narrow wheel that allows the robot to change the direction of an articulation with minimum friction. AZ1 can change direction using differential steering or by changing the direction of its articulations, making the platform omnidirectional (as represented in Fig. 2) and allowing it to move in tight areas (to handle straight or circular staircases in home environments). The robot can cross its articulations and lift itself up when it gets stuck over an obstacle. Being omnidirectional, it would also be possible for a group of AZ1 robots to change direction in a coordinated fashion while transporting together a common payload or large objects, without requiring a rotative fixture for the object to carry. Many other configurations can be imagined, and the 12 DOF on AZ1 give the robot great flexibility and versatility in its locomotion capabilities. Modularity is preserved at the structural level by putting the electrical and embedded systems inside the body of the robot, and by placing the actuators on the locomotion parts (i.e., Ferrite ServoDisc motors in the leg-track-wheel articulations), minimizing the size of the robot’s body and facilitating changes of the articulations. Equipped with its two packs of 24V Ni-MH cells, AZ1 had from 2.25 to 3.5 hours of

Fig. 2 AZIMUT omnidirectional capabilities.

Fig. 3 AZIMUT-2 subsystems.

?

?

?

?

harmonic drive reducer), providing propulsion to the robot in a wheel configuration or a track configuration, lighter and more powerful than the ones used in the first prototype. Figure 4 illustrates how the wheel-motor is reused in both the wheel and the track configurations. Increased stability and compliance by adding a vertical suspension and using an elastic actuator for the direction of AZIMUT’s articulations. An elastic element (e.g., a spring) is placed in the actuation mechanism and a sensor is used to measure its deformation, allowing to measure and control the forces at the actuator’s end [6], in order to improve robot motion over uneven terrains. The electrical and computing distributed embedded system has been improved. We now have small, stable and robust CAN modules (Direct Current (DC) Brush/Brushless 10 Amps motor drive; energy distribution; sonar sensor interface; LCD 128 64 bits monitor; wireless remote control) and I2C modules (DC 3 amps motor drive; infrared range sensor interface for stand-alone sensors). A Microchip PIC18F microcontroller board has been designed (and referred to as the PICoModul). We also have developed an innovative and efficient way to remotely program PICoModul over the CAN bus, allowing to easily monitor and change their configuration online, facilitating also maintenance. Visualizing state variables (values, graphs of temporal evolution) simplify calibration of the robot. AZ2 was designed to facilitate access to all its internal components. For instance, the connectors of the onboard computer were all made available outside the body of the robot, and the computing module can be easily detached from the base for software maintenance. AZ2 is smaller than AZ1, lighter (33.7 kg compared to 63.5 kg) and faster (directional speed of 180°/sec compared to 120°/sec, no load; maximum velocity of 1.4 m/s compared to 1.2 m/s).

III.

KINEMATIC ANALYSIS AND CONSTRAINTS

As a first object of study on AZ2, our interest is on its omnidirectional kinematic model, and more specifically on the AZIMUT wheel. The AZIMUT wheel is similar to a Castor wheel. The main difference consists in the orientation of the wheel plane. With a Castor wheel, the plane of the wheel contains also the steering axis, while the AZIMUT wheel plane is parallel to the steering axis but does not contain it. In addition, the steering axis and the wheel axis intersect in one point, as shown in Fig. 5. AZIMUT’s four steering axis of the wheels are located at the corners of a square. The steering angle range of each wheel is limited to 180 o . For example, wheel number 1 can steer from 45 o to 135 o . Its steering range comprises the full first quadrant and one half of the adjacent quadrants two and four, as illustrated in Fig. 6. Let us s define two reference frames: the global reference frame {O, X I ,YI } attached to the plane, and the local reference frame {P, X R ,YR } attached to the robot chassis (Fig. 7). Given the chassis velocity ˙I in the global reference frame, it is possible to represent the velocity ˙R in the local reference frame by using the well-known relationship between two reference frames [3]:

˙ = R( R
with:

)

˙

I

(1)

˙= x ˙

[

˙ y

˙

]

T

(2)

cos R( ) = sin 0

sin cos 0

0 0 1
(3)

These two research prototypes provide proof-of-concept demonstrations of a reconfigurable, modular and multi-modal locomotion robotic platform. AZ2 is the platform we use to conduct research and real world testing of advanced kinematic and locomotion control of mobile robots.

For each wheel, the rolling constraint enforces that all motion along the direction of the wheel plane must be accompanied by the appropriate amount of wheel spin ˙ so that there is pure rolling at the contact point. On the other hand, the non-conventional geometry of the steering mechanism has significant impact on this constraint. Steering must always be accompanied by the appropriate amount of wheel spin, as expressed by Equation (4):

Fig. 4 The wheel-motor is reused in both the wheel and the track configurations.

Fig. 5 AZIMUT wheel CAD drawings.

moves along a straight path with ICR at infinity. The measure of mobility above examines degrees of controllable freedom in relation to changes to wheel velocity. Steering can also have an impact on the chassis pose I . However, this impact is indirect because after changing the angles of the steerable AZIMUT wheels, the robot must move for the change in steering angle to have impact on pose. The number of independently controllable steering parameters (called degree of steerability) is given by: (4)
s

Fig. 6 Wheel quadrants and physical constraints.

[sin(

+

)

cos( +

)

d + l cos

]R( ) ˙I

r ˙ +d˙ =0

= rank(C)

(8)

Additionally, the sliding constraint enforces that the component of the wheel’s motion orthogonal to the wheel plane must be zero:

[cos(

+

)

sin( +

)

l sin

]R( ) ˙I = 0

(5)

Two cases can occur: If the four wheel’s axis does not intersect at the same point, the degree of steerability is not defined. 2. If the four wheel’s axis intersect at the same point, rank(C) = 2 and s = 2 . This result implies that the robot can place its ICR anywhere on the ground plane. 1. The degree of maneuverability is given by Equation (9). This means that the configuration space of AZ2’s chassis is 3.
M

The four sliding constraints of all wheels can be collected in the following matrix form:

C(

1,

2,

3,

4

)

R(

) ˙I

=0

(6) IV.

=

m

+

s

= 1+ 2 = 3

(9)

with C being a ( 4 3) matrix whose values depend from the steering angles i (t) . Knowing the number of independent constraints instantaneously, one can compute the robot’s degree of mobility [1]:
m

COMPUTING THE ADMISSIBLE VELOCITY SPACE

= 3 rank(C)

(7)

Two cases can occur: If the 4 wheel’s axis do not intersect at the same point, all the kinematical constraints are independent, so that rank(C) = 3 and m = 0 . In other words, the robot’s chassis is not be able to move. 2. If the four wheel’s axis intersect at the same point, 3 of 4 constraints become linearly dependant, implying that rank(C) = 2 and m = 1 . Therefore, the robot is able to turn around the point of intersection of the wheel’s axis. This point is called the instantaneous centre of rotation (ICR). When all of the wheel’s axis are parallel, the robot 1.

Our interest in this section is to study the different ways in which the robot can use its controllable degrees of freedom to position itself in the environment. Considering that the configuration space of the robot’s chassis is three, we will care about how the robot is able to move between various configurations. The computation of admissible velocity space is the first step in understanding how a robot can move from configurations space’s position A to position B. Using a method described in [4], we define a vector v containing: ? The linear velocities of the four wheel-ground contact points. ? The linear velocities of the four points situated on the four steering axis. ? The linear velocity of the chassis’s centre. ? The angular velocity of the chassis. ? Then angular velocities of the arms connecting the wheels. The rigid body constraint equations applied to the chassis and the four arms, plus the sliding constraint equations applied to the four wheels, can be written in a compact matrix form:

G v=0

(10)

Using linear algebra, it can be said that the null space G provides a base of the admissible velocity workspace. In
Fig. 7 Four wheeled AZIMUT-2

other words, any velocity state can be expressed by Equation (11):

v = null(G) p

(11)

Using this results: 1. If the wheel’s axis does not intersect at the same point, the dimension of the admissible velocity space is 4. The chassis cannot move while the four arms can move independently driven by parameter’s vector p[4 1] . 2. If the four wheel’s axis intersect at the same point, the dimension of the admissible velocity space is 5. The chassis can turn around the ICR and the four arms can move independently. The velocity vector is defined by choosing the parameters of p[5 1] . That means that ICR’s position relative to the chassis can be changed while the robot is moving. Fig. 8 illustrates the simulator implemented to visualize the velocity state vector graphically. This tool allows the user to define the free kinematical parameters and the ICR’s position as inputs and visualize the corresponding velocity state as output. V. EFFECTS OF THE STEERING ANGLES LIMITATIONS ON
THE VELOCITIES ADMISSIBLE SUBSPACES

Fig. 9 The seven different areas each corresponding to a locomotion mode. The shaded square represent the robot. When ICR is far from the robot center, the areas number 1 to 4 become negligible.

In theory, the same ICR position could be obtained with 2 4 = 16 combinations of arm placements. This is due to the fact that a given ICR position can be pointed by each arm with two theoretical steering angles, and +180 o . Due to the mechanical implementation of the AZIMUT wheel, some physical constraints have been added and most combinations become invalid. VI. IMPLEMENTATION OF THE KINEMATICAL MODEL TO CONTROL AZ2

We discovered that the resulting admissible velocity space consists of the addition of eight subspaces that touches each other in one point at the origin of the total space. One subspace corresponds to the steering reconfiguration mode while the others can be visualized graphically by decomposing the plane into seven areas (shown in Fig. 9). For each area, the ICR can be smoothly displaced without having to immobilize the robot. These seven zones define the seven locomotion modes of the robot, illustrated in Fig. 10 1) crab quadrant I; 2) crab quadrant IV; 3) crab quadrant III; 4) crab quadrant II; 5) tight turn; 6) car right-left; 7) car up-down.

First, let us describe the user interface implemented to control adequately the robot. The previous considerations about mobility of AZ2 justify two main operation modes: A) Reconfiguration mode in which all the steering angles are changed independently to reach a desired ICR. In this mode, the chassis does not move. B) Locomotion mode in which all the wheels points to the same ICR. In this mode, the chassis can move around the ICR. These two modes require the user to choose the position of desired ICR using the simulation interface shown in Fig. 8. We divide the plane into finite portions based on polar coordinates ( , ) . Additionally, our interface allows us to

Fig. 8 Simulation interface screenshot. User can select an ICR and the program computes the velocities of the wheels. Note the infinite outer ring that corresponds to straight motion of the chassis.

Fig. 10 Different positions of ICR and the associated possible movement of the robot. a) car up-down; b) car right-left, c) crab quadrant I; d,e) tight turn; f) reconfiguration mode

choose an infinite ICR corresponding to straight motion by selecting a point inside a special outer ring. The two modes of operation are: A) Reconfiguration mode. If two motors per wheel are used to motorize an AZIMUT wheel, then we are faced with a problem of hyper-motorization because steering involves only one degree of freedom per wheel. The principal problem due to hyper-motorization arises when the actuators use high impedance gearboxes as output stages. Small position/speed inconsistencies in regard to the kinematical constraints generate high internal forces/torques causing the actuators to fight each other, which can lead to potential damages. To solve this problem, one solution is to motorize the steering axis of steerable wheel with an on/off electromechanical brake [5]. The brake is controlled with a pulsed width modulation signal to allow the chassis to steer without immobilizing the robot. This results however in very slow rejection of external perturbations. Our solution consists of integrating a torsion spring between the two actuators. This is what is know to be serial elastic actuator [6]. By decoupling mechanically the two actuators with a spring, we are able to use a high ratio/high impedance gearbox as output stage without risk (Fig. 11). Measuring the spring angular displacement makes it possible to control the applied steering torque. We implemented the steering reconfiguration mode by setting a zero torque command to the steering actuator and a position/velocity command profile to the wheel actuator. As a result, the steering axis follows the wheel rotational displacement automatically. B) In locomotion mode, the user commands the robot to move by setting a motion parameter P . This parameter corresponds to a percentage of the fastest movement that the robot is able to perform around the user specified ICR, taking into account the speed saturation of the actuators that motorize the wheels. The vector mot containing the four rotational speeds associated to each wheel is calculated by using Equation (12):

mot

= L( , )

P 100

(12)

Our algorithm implementing Equation (12) uses a precomputed [4x1] vector L( , ) , which components are determined by four lookup tables. Each set of lookup table is different from one ICR position to another. The generation of the lookup tables was made automatically using a MATLAB computer program for each ICR position. VII. CONCLUSION AND FUTURE WORK

In this paper, we present an innovative wheeled mobile robot based on a new kind of wheel, the AZIMUT wheel. Using the wheel kinematical model, we demonstrate how to compute the mobility and steerability of the robot. We also demonstrate that AZIMUT is a non-holonomic omnidirectional robot. To overcome the hyper-motorisation issue, we propose a new design paradigm based on the utilization of four serial elastic actuators to motorize the steering. The controller implemented and based on the robot’s kinematical model shows excellent locomotion capabilities, allowing the robot to accelerate and decelerate around any ICR in a very stable manner despite the simplicity of the controller. The next step consists of implementing a controller making it possible to change the ICR without having to stop the robot. We also want to implement a feed-forward controller portion based on the knowledge of desired user acceleration and robot dynamical model. ACKNOWLEDGMENT F. Michaud holds the Canada Research Chair (CRC) in Mobile Robotics and Autonomous Intelligent Systems. This work is funded by the CRC, the Canada Foundation for Innovation and the Natural Sciences and Engineering Research Council of Canada. REFERENCES
[1] R. Siegwart, I. Nourbakhsh, Introduction to Autonomous Mobile Robots, MIT Press, 2004. [2] Michaud, F., Létourneau, D., Paré, J.-F., Legault, M.-A., Cadrin, R., rsenault, M., Bergeron, Y., Tremblay, M.-C., Gagnon, F., Millette, M., Lepage, P., Morin, Y., Caron, S. (2005), "Multi-modal locomotion robotic platform using leg-track-wheel articulations", Autonomous Robots, Special Issue on Unconventional Robotic Mobility, 18(2):137-156. [3] J.-J. Kraig, Introduction to Robotics, Mechanics and Control, Pearson Prentice Hall, 2005. [4] M. Lauria, S. Shooter, R. Siegwart, “Topological analysis of robotic Nwheeled ground vehicles,” in Proceedings of the 4th International Conference on Field and Service Robotics, July 14-16, 2003, Japan. [5] Y. Mori, E. Nakano, T. Takahahsi, “Mechanics, control and design methodology of the non-holonomic quasi-omnidirectional vehicle ODV9”, The International Journal of Robotic Research, 21(5-6): 511-525, MayJune 2002. [6] D. Robinson, “Design and analysis of series elasticity in closed loop actuator force control”, Ph.D. Thesis, Massachusetts Institute of Technology, 2000.

Fig. 11 AZIMUT-2 and AZIMUT wheel detailed mechanism


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