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SECTION–1
PLENARY PRESENTATIONS
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ABSTRACTIONS FOR LEGGED LOCOMOTION
GABRIEL A.D. LOPES
Delft Center for Systems and Control,
Delft Robotics Institute,
Delft University of Technology
Mekelweg 2, 2628CD, Delft, The Netherlands
∗ E-mail: G. A. DelgadoLopes@ tudelft. nl
www.dcsc.tudelft.nl:~glopes
The synchronization of legs in many-legged robots is a combinatorial problem that can be very critical for successful climbing or locomotion in very rough terrain. This paper addresses developing gait generation controllers with desirable synchronization properties by introducing a number of abstractions in the traditional modeling approaches. It describes how to use the max-plus algebra as a modeling tool for discrete-event systems and how it can be applied to legged locomotion.
Keywords: Legged locomotion, discrete event systems, max-plus algebra, robotics
1. Introduction
Legged robots offer far more versatile mobility than wheeled or tracked vehicles. The examples are everywhere in nature where animals are capable of negotiating a wide spectrum of terrain properties, such as inclination, viscosity, roughness. In natural human environments, populated with stairs, slopes, and ledges, legged robots have the potential to be the mobility platform of choice for applications such as home robotics, care and cure robots, agriculture, construction work, fire/police. The relevance of legged robots is already recognized in competitions where challenging locomotion environments are present, such as the Darpa Robotics Challenge. For all their virtues legs bring a large set of technological challenges. The most prominent arises from the impacts generated from intermittent interaction of legs with the ground on very dynamic robots. Leg impacts introduce a number of challenges from the mechanical engineering point of view where fatigue play a fundamental role. Recent developments to address impacts include the use of compliant legs1–3 and novel morphologies.4 From a systems and control perspective the main challenges arise in the complexity of the mathematical models. Systems with impacts result in hybrid dynamical models.5 For the case of legged robots intrinsic instability due to upright posture complicates even further the control design. The main control tasks for legged robots can be categorized as:
(1) Balancing – how to maintain an intrinsically unstable hybrid mechanical system in a stable upright configuration in the presence of disturbances or complex terrain.
(2) Synchronization – when multiple legs are present, in which order/configuration should these interact with the ground.
(3) Planning – deciding where and how to move.
(4) Optimization – how can motion be achieved to minimize cost of transport, maximize speed, etc.
This paper focuses on the 2nd challenge: the synchronization of legs in a many-legged robota. Section 2 briefly describes what is the synchronization problem and how it is traditionally approached in the scientific literature, both from the biology and robotics point of view. Current state-of-the-art control architectures for leg synchronization are presented. In Section 3 a different set of modeling and control tools are presented, called max-plus linear (MPL) systems. It is shown that these models are very suitable for legged locomotion. MPL systems are a class of discrete-event systems driven by the max-plus algebra.6 Using these tools a generic leg synchronization algorithm is presented in Section 4 as the solution to a mixed-integer linear programming problem. Section 5 describes explicit solutions for a class of gaits and Section 6 concludes with a list of extensions for the application of MPL control systems in legged locomotion. A brief description of the max plus algebra is given in Appendix A.
2. Leg synchronization
The term gait means a manner of walking. Human gaits include walking, running, and skipping. Quadrupedal animals exhibit many more gaits, including amble, trot, pace, canter, gallop, bound, etc. In the late eighteen hundreds Muybridge7 did pioneering work in photographing running horses and in the sixties Hildebrand8 presented a classification for gaits in terms of the stance time duty cycle and ph...
