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Research Papers

# Instant Center Based Kinematic Formulation for Planar Wheeled Platforms

[+] Author and Article Information
Amit Kulkarni1

Department of Mechanical Engineering, University of Texas at Austin, 1 University Station, R9925 Austin, TX 78712-1100akulkarni@mail.utexas.edu

Delbert Tesar

Department of Mechanical Engineering, University of Texas at Austin, 1 University Station, R9925 Austin, TX 78712-1100tesar@mail.utexas.edu

Table 2 presents the summary for up to the fifth order motion.

This is a brief list. A detailed review can be found in these papers.

Tesar provided notes at the workshop based on the lectures by Bottema.

Translated by Tesar in 1961 under an Army Research Office grant.

In case of a $J$ wheeled mobile platforms, $Ej$$(j=1,2,3,…,J)$ can represent $J$ wheel attachment points.

1

Corresponding author.

J. Mechanisms Robotics 2(3), 031015 (Jul 27, 2010) (12 pages) doi:10.1115/1.4001772 History: Received May 14, 2009; Revised March 09, 2010; Published July 27, 2010; Online July 27, 2010

## Abstract

For a general $J$ wheeled mobile platform capable of up to three-degrees-of-freedom planar motion, there are up to two $J$ independent input parameters yet the output of the platform is completely represented by three independent variables. This leads to an input parameter resolution problem based on operational criteria, which are in development just as they have been developed for $n$ input manipulator systems. To resolve these inputs into a meaningful decision structure means that all motions at the wheel attachment points must have clear physical meaning. To this effect, we propose a methodology for kinematic modeling of multiwheeled mobile platforms using instant centers to efficiently describe the motion of all system points up to the $nth$ order using a generalized algebraic formulation. This is achieved by using a series of instant centers (velocity, acceleration, jerk, and jerk derivative), where each point in the system has a motion property with its magnitude proportional to the radial distance of the point from the associated instant center and at a constant angle relative to that radius. The method of instant center provides a straightforward and physically intuitive way to synthesize a general order planar motion of mobile platforms. It is shown that a general order motion property of any point on a rigid body follows two properties, namely, directionality and proportionality, with respect to the corresponding instant center. The formulation presents a concise expression for a general order motion property of a general point on the rigid body with the magnitude and direction separated and identified. The results are summarized for up to the fifth order motion in the summary table. Based on the initial formulation, we propose the development of operational criteria using higher order properties to efficiently synthesize the motion of a $J$ wheeled mobile platform.

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## Figures

Figure 1

A generalized notation describing the kth order, three-DOF planar motion of a rigid body

Figure 2

The first order motion description of a general point E using a body fixed frame at the velocity IC

Figure 3

First order motion description of various points in the body using the velocity IC

Figure 4

Second order motion description of various points in the body using the acceleration IC

Figure 5

Tangential and normal components of the acceleration of a general point E with respect to the acceleration IC

Figure 6

Third and fourth order motion descriptions using the corresponding ICs

Figure 7

kth order kinematics of a general rigid body

Figure 8

Mobile platform traversing an S shaped trajectory

Figure 9

Higher order motion planning for the mobile platform traversing the trajectory

Figure 10

Special case for the first order, three-DOF planar motion of a rigid body when the velocity of point P is zero. The location of the velocity IC is coincident with point P.

Figure 11

Special case for the first order, three-DOF planar motion of a rigid body when the angular velocity of the body is zero. The location of the velocity IC is at the infinity.

Figure 12

The location of the velocity IC for a two wheeled differentially driven platform is constrained to the axis of wheel rotation

Figure 13

The velocity IC coincident with one of the wheel center for a two wheeled differentially driven platform

Figure 14

Special case for the second order, three-DOF planar motion of a rigid body when the angular velocity of the body is zero

Figure 15

Special case for the second order, three-DOF planar

Figure 16

Special case for the second order, three-DOF planar motion of a rigid body when the acceleration of point p is zero

Figure 17

Special case for the second order, three-DOF,planar motion of a rigid body when the angular velocity and the angular acceleration of the body is zero

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