Maximizing Walking Step Length for a near Omni-directional Hexapod Robot

A foot path planning algorithm is presented for a robot with six limbs symmetrically located on the faces of its hexagonal body, enabling it to walk at a constant height with an alternating tripod gait. The symmetry results in near omni-directional locomotion capability, so the algorithm is formulated for walking in any direction and at any height. The approach is to determine the maximum length foot path through each limb’s workspace and then modify those foot paths based upon static stability analysis. The stability analysis is conducted in two phases to ensure stability without excessively reducing step length. Compared to an optimization approach, the algorithm yields foot paths within 9.1% of the maximal foot paths for all directions and heights. Unlike the optimization approach, the developed algorithm is computationally efficient enough to be implemented in realtime. INTRODUCTION The majority of hexapod robots constructed to date have had an insect-like structure with the limbs located laterally on the body in three opposing pairs. Examples range from the Adapative Suspension Vehicle (ASV) [1], a hydraulically actuated machine in excess of three metric tons that was capable of walking with several different gaits and carrying an operator, to RHex [2], a 7 kg electrically actuated machine that is capable of various running gaits. Both of these robots were operated on uneven terrain, highlighting the potential benefit of limbed machines compared with conventional wheeled and tracked vehicles. Beyond the potential to negotiate natural terrain, though, limbed machines also offer the advantage of using their limbs for multiple functions. In fact, this paper intentionally uses “limb” in place of the more common “leg” to emphasize the multifunctionality. A hexapod, for example, might employ all of its six limbs to traverse a particularly difficult stretch of terrain and then use only two of its limbs to dig a hole or manipulate an object while standing on its other four limbs. The lateral positioning of the limbs for insect-like hexapods favors the function of locomotion over that of manipulation. Note that the limbs of both the ASV and RHex were designed exclusively for locomotion, providing large stroke in the direction of the body heading. The LEMUR I [3] hexapod has the same insect-like structure, but its two front limbs are different from the other four, having one additional degree of freedom (DOF) specifically to enable manipulation. Specialization of only the front limbs for manipulation can endow a limbed robot with some of the advantages of multifunctional limbs, but it may not be adequate in extreme cases. In highly unstructured terrain or confined spaces, such as disaster rubble, a limbed robot may not have enough room to maneuver its body into a position from which the manipulation-specialized limbs can reach targets of interest. The Jet Propulsion Laboratory’s LEMUR II [4] hexapod shown in Figure 1 addresses this limitation with six identical limbs located symmetrically around its hexagonal body, all having four DOF’s to facilitate manipulation. Symmetrical arrangement of a hexapod’s limbs was previously employed in Odex 1 [5], Arai’s limb-mechanism robot [6], 1 Copyright  2004 by ASME Figure 1. LEMUR II HEXAPOD WITH ONE LIMB HAVING A TOOL IN PLACE OF THE STANDARD LOWER SEGMENT. and Nataraj [7]. LEMUR II’s on-board sensing, a pair of stereo cameras, are mounted on a turn table that can rotate nearly 360 degrees. As a result, the robot can orient its vision system toward any target and then manipulate that target with its nearest limb without changing its body orientation. LEMUR II’s symmetrical limb configuration makes it a near omni-directional hexapod. The qualifier “near” indicates that the robot cannot truly move in every direction with equal ease due to the workspace limitations of the limbs. This point is developed in detail in the paper. Still, LEMUR II’s normal walking mode requires no turning of the body, only changes in directional heading. Since the limbs are neither specialized for nor optimally arranged for locomotion, care is needed in developing the walking algoithms so that enabling manipulation does not unacceptably diminish mobility. This paper presents a foot path planning algorithm that can be implemented in real time on a robot of LEMUR II’s structure to maximize the step length in walking at a constant height over level terrain with a statically stable alternating tripod gait. This algorithm represents a first step toward more generalized gait planning for robots like LEMUR II that are not specialized exclusively for locomotion. Since LEMUR II was designed for operation in microgravity, dynamic effects are not of immediate interest, but rather a subject of future work. In the following section, the LEMUR II robot is described in greater detail, and then, the foot path planning algorithm is presented. Maximizing the step length in an alternating tripod gait is subsequently formulated as a nonlinear optimization problem to determine the true maximum. Finally, the results of the developed foot path planning algorithm are compared with those of the optimization approach to quantify the performance at various walking heights and in various walking directions. ROBOT DESCRIPTION The LEMUR II hexapod shown in Figure 1 was designed for the performance of assembly, inspection, and maintenance tasks at space installations [4]. Thus, operational flexibilty motivated the development of multi-functional limbs. Each of the six limbs, arranged symmetrically on the hexagonal body, has four DOF’s and includes a quick-connect end-effector below the distal joint that enables rapid change-out of various tools. The 1-DOF distal joint is a revolute, corresponding roughly to a knee or elbow joint. The other three DOF’s arise from three revolute joints whose axes intersect orthogonally at a single point near where the limb is attached to the body. Since this configuration is kinematically equivalent to a spherical joint, it will be referred to simply as a single proximal joint having three DOF’s. The proximal joint corresponds roughly to a hip or shoulder joint. The use of “distal” and “proximal” in place of “hip” and “knee” or “shoulder” and “elbow” again is meant to emphasize that the limbs function both as arms for manipulation and legs for locomotion, although locomotion is the focus of this work. Each of the revolute joints is directly driven by an electric gearmotor and harmonic drive combination. The modified Denavit-Hartenberg [8] parameters for one limb using Craig’s [9] convention for assigning coordinate systems are given in Table 1, and Figure 2 is the corresponding kinematic sketch of a single limb in its zero position. The length of the upper limb segment is Lu, and the length of the lower limb segment is Ll . Figure 3 is a top view of the body that shows the coordinate system of the second revolute joint for each of the six limbs. Leading numerical superscripts indicate the limb number, and the “B” superscript indicates the body-fixed coordinate system. The length of each side of the body is Lb. Note that when a limb is in its zero position as shown in Figure 2, it is parallel to the face of the body to which it is attached. Furthermore, the first revolute joint in each limb provides for the roll motion of the entire limb relative to the body, which is necessary for manipulation Table 1. MODIFIED DENAVIT-HARTENBERG PARAMETERS.