Articulated Wheeled Robots: Exploiting Reconfigurability and Redundancy

Articulated Wheeled Robotic (AWR) locomotion systems consist of chassis connected to a set of wheels through articulated linkages. Such articulated “leg-wheel systems” facilitate reconfigurability that has significant applications in many arenas, but also engender constraints that make the design, analysis and control difficult. We will study this class of systems in the context of design, analysis and control of a novel planar reconfigurable omnidirectional wheeled mobile platform. We first extend a twist based modeling approach to this class of AWRs. Our systematic symbolic implementation allows for rapid formulation of kinematic models for the general class of AWR. Two kinematic control schemes are developed which coordinate the motion of the articulated legs and wheels and resolve redundancy. Simulation results are presented to validate the control algorithm that can move the robot from one configuration to another while following a reference path. The development of two generations of prototypes is also presented briefly. INTRODUCTION In recent times, a new class of robotic locomotion systems – articulated wheeled robot (AWR) – consisting of a main chassis connected to a set of wheels with ground contact via articulated chains have been proposed. This class of so called ‘leg-wheeled’ systems has been getting considerable attentions due to their advantages over traditional wheeled systems and legged systems in various applications as planetary explorations [1, 2], agriculture [3], rescue operations and wheelchairs [4]. Adding articulations between the wheels and chassis allows the wheel placement with respect to chassis to change during locomotion either passively or actively, thus AWRs can be briefly divided into these two categories. The main research in passive AWRs concerns designing suspension mechanism to negotiate with the uneven terrain. The planetary rovers [1] developed at Jet Propulsion Laboratory (JPL) and the Shrimp rover [2] have shown enhanced terrain adaptability featuring novel suspension design such as rocker-bogie and fourbar mechanism. They change their configuration according to the changing terrain topology. Passive AWRs are usually designed to have fewer degrees of freedom (DOFs) such that the weight of the system can be supported by the structure. The main advantages of passive AWRs are in terms of power consumption, payload capacity, and controller design. Active articulations further enhance the mobility of the robots to obtain better performance, such as stability and traction. They have been demonstrated by sample return rover (SRR) [5], ATHLETE rover [6], WAAV [7], Workpartner [3], Hylos [8], and variable footprint wheel chair [4]. The redundant actuated DOFs bring the system capability to optimize certain performance index such as stability. On the other hand, more actuators add extra weight and control complexity. In most applications, the wheel of AWRs is considered as a rigid disk with a single point of contact with the terrain surface. This means that the motion of the wheel is restricted by nonholonomic constraints. These constraints could be violated with slipping and skidding which are main sources of large energy consumption and measurement uncertainty. Minimization of slipping and skidding is usually desired and can be achieved either by a good kinematic design or proper cooperation of the rolling or steering of the wheels. Holonomic constraints in the articulations also increase the complexity of the system for people to relate the motion between wheels and chassis. Thus, the design, navigation and control of AWRs require a general framework for systematic kinematic modeling and analysis. Kinematic modeling of ordinary wheeled robots (OWRs, which can be seen as a subset of AWRs) has been dealt extensively. Muir and Newman [9] derived the equation of motion of OWRs using matrix transformation. Campion et al. [10] classified OWRs based on kinematic models developed using vector approach and nonholonomic constraints. Yi and Kim [11] presented modeling of omnidirectional wheeled robots with slipping. Fewer efforts have been focusing on AWRs. Grand et al. [8] presented a general geometric modeling approach and controlled the locomotion and posture separately. Tarokh and McDermott [12] used symbolic derivatives of transformation Proceedings of DSCC2008 2008 ASME Dynamic Systems and Control Conference October 20-22, 2008, Ann Arbor, Michigan, USA 1 Copyright © 2008 by ASME DSCC2008-2193 matrices with consideration of wheel slip and discussed three different kinematic forms for passive rovers. Choi and Sreenivasan [13] construct the kinematic model using screws and proposed a force distribution algorithm. Twist based approaches have been used to analyze motion and force capabilities systematically in other contexts, such as for parallel manipulators or multifinger grasping. However such methods have not been reported for modeling of AWRs, and especially articulated systems with rolling wheel ground contacts. Thus, one of the principal contributions of this paper is to extend the systematic twist-based modeling framework to this class of AWRs. Further, automating this process by using the symbolic toolbox in MATLAB, facilitates the rapid modeling and analysis of any given design of an AWR. We will also illustrate our modeling process in the context of design and analysis of a novel articulated omnidirectional robot – the ROAMeR. Traditional planar wheeled platforms in indoor application driven by fixed or centered orientable wheels are subjected to nonholonomic constraints which restricts the motion of the overall system [10]. In practice, such vehicles are typically unable to move their payload in all directions with equal ease. Hence there has been an increased interest in developing wheeled platform with omnidirectional motion capability. Many omnidirectional vehicle designs have been proposed using Mecanum/Swedish or Ball wheels. Wada and Asada [4] built a reconfigurable wheelchair using ball wheels. Song and Byun [14] presented a robot with steerable omni-directional wheels. Such systems face many challenges including discontinuous ground contact, poor ground clearance, or complicated mechanical design. Hence an exclusively disk-wheeled based design is preferred from the view point of ease of actuation and robustness. However, additional articulations need to be introduced in order to allow for adequate mobility within the system. Caster wheels have been implemented as the simplest articulated wheel for allowing omnidirectional mobility. For example, Yi and Kim [11] and Holmberg and Khatib [15] built and analyzed omnidirectional robots driven by powered caster wheels – however, the locations of the caster wheel w.r.t the chassis are always constant. The presence of more articulations within the leg-wheel chain further provides reconfigurability by allowing relocation of the wheel with respect to the chassis. There are many scenarios where planar AWRs could benefit from reconfigurability (which in the past has often only been explored in the context of uneven terrain locomotion). For instance, the robot base may need to be compact when passing a narrow doorway and be extended to enhance stability when manipulating heavy objects. Hence in this paper we examine a wheeled platform design (with active articulations and actively driven disk wheels) for the purpose of achieving omnidirectional mobility together with the ability to reconfigure for different tasks. The modeling and control complexity of the ROAMeR increase with the addition of these articulations and their interaction with the contact constraints. We will address the modeling within the twist based framework leading up to development of 2 kinematic control laws for our ROAMeR. The focus of these laws is on resolving the redundancy while allowing for simultaneous trajectory tracking and configuration control of the ROAMeR. The rest of the paper is organized as follows: Section 2 discusses twist based modeling. Modeling of a planar reconfigurable omnidirectional robot is presented in Section 3. In Section 4, two kinematic control schemes are proposed. Section 5 discusses the simulation result, followed by prototypes in Section 6. Section 7 concludes the paper. TWIST BASED KINEMATIC MODELING As we are focusing on AWRs, we will not discuss the cases where the robot has contact points to the ground that is not on the wheel. To establish the kinematic model that relates the motion of the robot body and the motion of the wheels and linkages, we will first define frames of reference properly, then find the twists expressed in a sequence of local frames starting from body fixed frame. By appropriate transformation, we can express any twist in one single frame and assemble them as the Jacobian matrix of the robot. A general model of AWR is shown in Fig. 1, we define an inertial frame of reference{ } ( , , , ) f F O X Y Z = , and at any time, the robot has an instantaneous frame{ } ( , , , ) b x y z B O b b b = attached to its body that moves with the robot, where b O is the point of interest on the robot (Center of Mass is often chosen). The configuration of the main body could be defined as [ ] T x y z φ θ ψ with respect to the inertial frame. The robot could possess n branches. Each of them consists of any number of linkages and end with one wheel. Each wheel has a coordinate frame { } ( , , , ) w x y z W O w w w = attached to the wheel axle (for simplicity, we will neglect the subscript i for labeling branch), w O is the center of the wheel and z w lies on the wheel axle. The dashed line in the figure between the chassis and the wheel represents any set of links and joints that exists between these two frames, including the steering and suspension mechanism. We define 0 B A A the transformation between body frame and joint 1 frame, 1 , 1,2, 1 j j A j m − = − the transformation between joint j and joint 1 j + frame, m W A the transformation between joint m frame and wheel frame. These transformations can be expressed easily using D-H parameters. FIGURE 1: GENERAL ARTICULATED WHEEL ROBOT 2 Copyright © 2008 by ASME In our model, each wheel is assumed to be represented by a rigid disc with a single point of contact with the terrain surface. The wheel plane is defined at the center of the wheel and perpendicular to the wheel axle ( z w ), that is, the plane formed by x w and y w .and a tangent contact plane is defined perpendicular to the wheel plane at the contact point, as shown in Fig. 2. FIGURE 2: RIGID DISK WHEEL GROUND CONTACT MODEL Multiple contact points on wheel offer a challenge for analysis and control and we will not consider it for the present. In the real application it is also quite difficult to sense or estimate the position of multiple contact points. For the single point contact, a contact frame { } ( , , , ) c x y z C O c c c = is defined as the contact frame assigned at each wheel’s contact point as illustrated in Fig. 2, where z c is a unit vector in the wheel plane and normal to the tangent contact plane at the point of contact, that is, a vector point to the center of the wheel. x z z c c w = × , and y z x c c c = × . γ is the contact angle defined as the angle between x c and x w , since x c is always lies in the wheel plane. This angle varies when the AWR is moving on uneven terrain or performing reconfiguration. It can be measured using force sensor on the wheel axle (or estimated). The transformation between the wheel frame and contact frame is denoted by the homogeneous transformation cos 0 sin sin sin 0 cos cos 0 1 0 0 0 0 0 1 W C r r A γ γ γ γ γ γ − ⎡ ⎤ ⎢ ⎥ − ⎢ ⎥ = ⎢ ⎥ − ⎢ ⎥ ⎣ ⎦ (1) The homogeneous transformation between any two frames can be obtained by matrix multiplication and inversion. For instance, B W A can be written as successive multiplication of several homogeneous transformations 0 0 1 B B A Am W A A W A A A A = (2) and overall transformation from body frame to contact frame can be found as B B W C W C A A A = (3) We define the velocity of the main body with respect to inertial frame expressed in body frame as B F B V ⎡ ⎤ ⎣ ⎦ . The velocity of contact frame with respect to inertial frame expressed in contact frame is defined as C F C V ⎡ ⎤ ⎣ ⎦ . In order to find the 6 dimensional twist vector representing joint velocity in local frame b a b V ⎡ ⎤ ⎣ ⎦ , one can either write it out directly (it is usually a simple motion for a single joint) or compute it by first finding the twist matrix and extract out the twist vector. The twist matrix is given by 3 3 3 1 1 0 0 b a a a T b b b v T A A × × − Ω ⎡ ⎤ ⎡ ⎤ = = ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ i (4) where a b A is the homogeneous transformation between two frames a and b , [ ] 3 1 1 2 3 T v v v v × = is the translational velocity and Ω is a 3 3 × skew symmetric matrix that represents the angular velocity of frame b with respect to frame a expressed in b