Kinematics of Spherical Robots Rolling over 3D Terrains

Abstract

Although the kinematics and dynamics of spherical robots (SRs) on flat horizontal and inclined 2D surfaces are thoroughly investigated, their rolling behavior on generic 3D terrains has remained unexplored. This paper derives the kinematics equations of the most common SR configurations rolling over 3D surfaces.First, the kinematics equations for a geometrical sphere rolling over a 3D surface are derived along with the characterization of the modeling method. Next, a brief review of current mechanical configurations of SRs is presented as well as a novel classification for SRs based on their kinematics. Then, considering the mechanical constraints of each category, the kinematics equations for each group of SRs are derived.Afterwards, a path tracking method is utilized for a desired 3D trajectory. Finally, simulations are carried out to validate the developed models and the effectiveness of the proposed control scheme. climb obstacles, assuming the condition to be static. From a different view, rolling of SRs are studied where the desired path is assumed to be a straight line with constant slope or a single step obstacle [9-11] and a 2D curved path with variable-slope [12] respectively. In [13], authors have investigated dynamics of Martian tumbleweed rovers while this special type of SR rolls in its heading direction and the turning action is not considered for them. In fact, while several researches have been done on 3D kinematics of other types of mobile robots such as legged [14] and wheeled robots [15,16], to the best of the authors' knowledge, the general problem of kinematics of SRs rolling on 3D terrains has not been investigated in the literature. The motivation to address this problem is that, while many applications of the SRs are on flat surfaces such as indoor [17], and paved roads [18], for a variety of applications such as agriculture [19], surveillance [20], environmental monitoring [21], and even planetary explorations [22], they would get exposed to uneven terrains.In this work, prior to deriving the kinematics of SRs on 3D terrains, a general method for modeling a geometrical sphere rolling over a mathematically known 3D surface is developed. Then, the derived equations are expanded in order to be applied to SRs considering their specifications. Concretely, a variety of mechanisms are utilized in SRs to provide the required propelling torques and forces for their rolling action. Each configuration imposes its own kinematical constraints on the SRs' rolling motion. Therefore, to study the kinematics of SRs it is essential to classify current and feasible designs of SRs accordingly.There are a few SR classifications available in the literature. In a survey [6], SRs are classified based on their mechanical driving principles as: 1) Barycenter offset (BCO), 2) Conservation of angular momentum (COAM), and 3) Shell transformation (OST). In another review, SRs are classified based on their mechanical configurations [23], e.g., single wheel, hamster wheel, pendulum driven, gimbal mechanism, single ball...

Design and implementation of a maxi-sized mobile robot (Karo) for rescue missions

“…Since each of DC motors has a unique mechanical specification (i.e., torque constant and nominal current), the corresponding load torque at each measured current value is calculated by using Eq. (11), taken from the Maxon Motor datasheet, to verify that both motors operate within the suggested continuous range by the manufacturer company. The torque constant K m is 38.5 mNm/A and 30.2 mNm/A, and the nominal current I 0 is 236 mA and 137 mA for the 200 W and 150 W DC motors, respectively.…”

“…However, there are several possibilities: the slippage between tracks and the floor; uneven wear on tracks; and possible shocks during the transition between surfaces. Although some of the measurements pass the 10.8 A (405 mNm) maximum continuous current (torque) of the 200 W motor, the overall performance (11) does not exceed the continuous operation range, so the robot can smoothly operate in various conditions. This can be seen clearly from the 8 A difference between the measured current on a flat surface and an inclined one.…”

“…However, the mobile platforms introduced in [7] and [8] can also be known as two early experimental efforts in design and implementation of mobile robot platforms which were potentially suited for USAR, though the authors did not explicitly mention their USAR applications. Later on, rescue robotics started getting more attentions gradually among the mobile robotics research community [9], which led to developing rescue robots with a variety of locomotion mechanisms such as spherical [10,11], legged [12], wheeled [13], and tracked [14] locomotion.…”

Design, Modelling and Control of SPIROS: The Six Propellers and Intermeshing Rotors Based Omnidirectional Spherical Robot

“…We have described the mechanical design of the intermeshed rotary thrusters, the outer gridshell and the full robot assembly in detail. Also, as per the proposed classification and the constraints of SRs in [25], the dynamics equations for SRs rolling over 3D terrains are derived. A modified pure pursuit method [26] is utilized for the path tracking control problem of derived models.…”

“…4) The moments generated and the frictional force are always enough to provide the required driving force to the robot. Regarding to the kinematics equations derived by Moazami et al [25] based on the different categories of the SRs rolling over the 3D terrains, the transformation between the world frame W co-ordinates and the tangent plane frame T coordinates is given by:…”