Instant center based kinematic and dynamic motion synthesis for planar mobile platforms
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For a general J wheeled mobile platform capable of up to 3-Degrees-Of-Freedom (DOF) planar motion, there are up to 2J independent input parameters yet the output of the planar platform is specified with only three independent parameters. Currently, the motion synthesis for such platforms is done with a Jacobian based “pseudo” inverse that uses a rectangular matrix for Jacobian. However, a mobile platform is a parallel mechanism and has a more direct solution to the inverse kinematics problem. To this effect, we propose a physical methodology for kinematic modeling of multi-wheeled mobile platforms using Instant Centers (IC) to describe the kinematic state of all system points up to the kth order using a generalized algebraic formulation. This is achieved by using a series of ICs (velocity, acceleration, jerk, etc.) where each point in the system has a time state with its magnitude proportional to the radial distance of the point from the associated IC and at a constant angle relative to that radius. The use of IC’s for mobile platform kinematics is not new, however we present a completely generalized and extensive formulation that also treats the higher order kinematics. To the best of our knowledge, this is the first time the third and higher order ICs have been presented in the literature. The components of this research effort are: (i) extension of the theory of instantaneous invariants to the higher order motion by generalizing the theory to any order, (ii) studying some special case 1-DOF, 2-DOF motions to understand the physical nature of the higher order ICs, (iii) applying the results of (i) and (ii) to the motion synthesis of planar, wheeled mobile platforms by first categorizing them into four distinct categories, and (iv) studying the dynamic model of a representative mobile platform to emphasize the importance of wheel dynamics and traction parameters on the performance of the mobile platform. The IC based formulation presents a concise expression for a general order time state of a general point on the rigid body with the magnitude and direction separated and identified. We showed that the method based on instant centers provides a straightforward and yet physically intuitive way to synthesize a general kth order planar motion of mobile platforms. The study of special case 1-DOF/2-DOF motions emphasized the geometric nature of the higher order ICs and also helped understand the influence of instantaneous kinematic states (such as angular velocity _, angular acceleration, _, etc.) on the various ICs. The application of this theory to planar mobile platform allowed us to categorize the platforms based on their dexterity and to generalize the motion synthesis to some extent. The study of the dynamic model of a representative mobile platform showed us that the redundant inputs (2J inputs versus 3 outputs) in this case may be employed to sustain and manage the uncertainties and nonlinearities in the wheel ground interaction.