Triple Stance Phase Displacement Analysis With Redundant and Nonredundant Sensing in a Novel Three-Legged Mobile Robot Using Parallel Kinematics

Abstract

This paper presents the forward and inverse displacement analysis of a novel three-legged walking robot Self-excited Tripedal Dynamic Experimental Robot (STriDER) in its triple stance phase. STriDER utilizes the concept of passive dynamic locomotion to walk, but when all three feet of the robot are on the ground, the kinematic configuration of the robot behaves like an in-parallel manipulator. To plan and control its change in posture, the kinematics of its forward and inverse displacement must be analyzed. First, the concept of this novel walking robot and its unique tripedal gait are discussed, followed by the overall kinematic configuration and definitions of its coordinate frames. When all three feet of the robot are on the ground, by assuming there is no slipping at the feet, each foot contact point is treated as a passive spherical joint. Kinematic analysis methods for in-parallel manipulators are briefly reviewed and adopted for the forward and inverse displacement analysis for this mobile robot. Both loop-closure equations based on geometric constraints and the intersection of the loci of the feet are utilized to solve the forward displacement problem. Analytical solutions are identified and discussed in the cases of redundant sensing with displacement information from nine, eight, and seven joint angle sensors. For the nonredundant sensing case using information from six joint angle sensors, it is shown that analytical solutions can only be obtained when the displacement information is available from unequally distributed joint angle sensors among the three legs. As for the case when joint angle sensors are equally distributed among the three legs, it will result in a 16th-order polynomial with respect to a single variable, and closed-form forward displacement solutions can be obtained. Finally, results from the simulations are presented for both inverse displacement analysis and the nonredundant sensing case with equally distributed joint angle sensors. It is found that at most 16 forward displacement solutions exist if displacement information from two joint angle sensors per leg are used and one is not used.

Closed-Form Solutions to the Kinematics of a Parallel Locomotion Mechanism With Actuated Spoke Wheels

“…In Eqs. (1)(2)(3), l a is the length of the axle; θ is the angular displacement of the two spoke wheels; d 1 and d 2 are the linear displacements of the left and right contacting spokes respectively, measured from the centers of the wheels to the contact points. d 1 and d 2 must comply with the following constraint:…”

“…Repeating Eqs. (1)(2)(3)(4)(5)(6)(7)(8), the information about the body's new configuration can be updated based on new joint displacements.…”

“…For such locomotion mechanisms, a fundamental research on their kinematics is quite necessary, as it will lay the foundation for other studies such as design optimization, dynamics modeling, nonlinear control, motion planning and so on. [3][4][5][6][7][8]. The first robot STriDER (Self-excited Tripedal Dynamic Experimental Robot) is a three-legged robot that utilizes its built-in passive dynamics for walking.…”

“…The kinematics in its triple stance phase has been investigated in Refs. [3]. The second robot IMPASS (Intelligent Mobility Platform with Active Spoke System) is an actuated spoke wheeled robot that has various topologies with respect to the ground.…”

Solution of underdetermined systems of equations with gridded a priori constraints

“…In the case of various non-linear problems, such as those arising from observations of distances and angles, as in various fields of Geodesy, algebraic solutions are obtained on the basis of linearization of the observation equations leading to a system of equations (Mikhail 1976 ). In the case of highly non-linear systems, however, this is not possible, and either certain observation equations are selected to solve a non-redundant system (Ren and Hong 2009 ), or various numerical/statistical, usually Monte Carlo-based approaches (or genetic algorithms, especially PSO and annealing simulations, Pedersen et al 2003 ; Li 2009 ; Voglis et al 2012 ) are used. Some limitations of these techniques are that they usually ignore the error properties (uncertainties) of observations and of the solutions, and they may be trapped in local solutions (see Saltogianni and Stiros 2012a ).…”